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Quadratic equation
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This article is about algebraic equations of degree two and their solutions. For equations of degree four, see Quartic equation. For functions defined by a polynomial of degree two, see Quadratic function. For the case of more than one variable, see Conic section and Quadratic form.
The quadratic formula for the roots of the general quadratic equation
In algebra, a quadratic equation (from the Latin quadratus for "square") is any equation having the form
a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0}
where x represents an unknown, and a, b, and c represent known numbers such that a is not equal to 0. If a = 0, then the equation is linear, not quadratic. The numbers a, b, and c are the coefficients of the equation, and may be distinguished by calling them, respectively, the quadratic coefficient, the linear coefficient and the constant or free term.[1]
Because the quadratic equation involves only one unknown, it is called "univariate". The quadratic equation only contains powers of x that are non-negative integers, and therefore it is a polynomial equation, and in particular it is a second degree polynomial equation since the greatest power is two.
Quadratic equations can be solved by a process known in American English as factoring and in other varieties of English as factorising, by completing the square, by using the quadratic formula, or by graphing. Solutions to problems equivalent to the quadratic equation were known as early as 2000 BC.
Contents
[hide]
1 Solving the quadratic equation
1.1 Factoring by inspection
1.2 Completing the square
1.3 Quadratic formula and its derivation
1.4 Reduced quadratic equation
1.5 Discriminant
1.6 Geometric interpretation
1.7 Quadratic factorization
1.8 Graphing for real roots
1.9 Avoiding loss of significance
2 Examples and applications
3 History
4 Advanced topics
4.1 Alternative methods of root calculation
4.1.1 Vieta's formulas
4.1.2 Trigonometric solution
4.1.3 Solution for complex roots in polar coordinates
4.1.4 Geometric solution
4.2 Generalization of quadratic equation
4.2.1 Characteristic 2
5 See also
6 References
7 External links
Solving the quadratic equation[edit]
Figure 1. Plots of quadratic function y = ax2 + bx + c, varying each coefficient separately while the other coefficients are fixed (at values a = 1, b = 0, c = 0)
A quadratic equation with real or complex coefficients has two solutions, called roots. These two solutions may or may not be distinct, and they may or may not be real.
Factoring by inspection[edit]
It may be possible to express a quadratic equation ax2 + bx + c = 0 as a product (px + q)(rx + s) = 0. In some cases, it is possible, by simple inspection, to determine values of p, q, r, and s that make the two forms equivalent to one another. If the quadratic equation is written in the second form, then the "Zero Factor Property" states that the quadratic equation is satisfied if px + q = 0 or rx + s = 0. Solving these two linear equations provides the roots of the quadratic.
For most students, factoring by inspection is the first method of solving quadratic equations to which they are exposed.[2]:202–207 If one is given a quadratic equation in the form x2 + bx + c = 0, the sought factorization has the form (x + q)(x + s), and one has to find two numbers q and s that add up to b and whose product is c (this is sometimes called "Vieta's rule"[3] and is related to Vieta's formulas). As an example, x2 + 5x + 6 factors as (x + 3)(x + 2). The more general case where a does not equal 1 can require a considerable effort in trial and error guess-and-check, assuming that it can be factored at all by inspection.
Except for special cases such as where b = 0 or c = 0, factoring by inspection only works for quadratic equations that have rational roots. This means that the great majority of quadratic equations that arise in practical applications cannot be solved by factoring by inspection.[2]:207
Completing the square[edit]
Main article: Completing the square
Figure 2. For the quadratic function y = x2 − x − 2, the points where the graph crosses the x-axis, x = −1 and x = 2, are the solutions of the quadratic equation x2 − x − 2 = 0.
The process of completing the square makes use of the algebraic identity
x 2 + 2 h x + h 2 = ( x + h ) 2 , {\displaystyle x^{2}+2hx+h^{2}=(x+h)^{2},}
which represents a well-defined algorithm that can be used to solve any quadratic equation.[2]:207 Starting with a quadratic equation in standard form, ax2 + bx + c = 0
Divide each side by a, the coefficient of the squared term.
Subtract the constant term c/a from both sides.
Add the square of one-half of b/a, the coefficient of x, to both sides. This "completes the square", converting the left side into a perfect square.
Write the left side as a square and simplify the right side if necessary.
Produce two linear equations by equating the square root of the left side with the positive and negative square roots of the right side.
Solve the two linear equations.
We illustrate use of this algorithm by solving 2x2 + 4x − 4 = 0
1 ) x 2 + 2 x − 2 = 0 {\displaystyle 1)\ x^{2}+2x-2=0}
2 ) x 2 + 2 x = 2 {\displaystyle 2)\ x^{2}+2x=2}
3 ) x 2 + 2 x + 1 = 2 + 1 {\displaystyle 3)\ x^{2}+2x+1=2+1}
4 ) ( x + 1 ) 2 = 3 {\displaystyle 4)\ \left(x+1\right)^{2}=3}
5 ) x + 1 = ± 3 {\displaystyle 5)\ x+1=\pm {\sqrt {3}}}
6 ) x = − 1 ± 3 {\displaystyle 6)\ x=-1\pm {\sqrt {3}}}
The plus-minus symbol "±" indicates that both x = −1 + √3 and x = −1 − √3 are solutions of the quadratic equation.[4]
Quadratic formula and its derivation[edit]
Main article: Quadratic formula
Completing the square can be used to derive a general formula for solving quadratic equations, called the quadratic formula.[5] The mathematical proof will now be briefly summarized.[6] It can easily be seen, by polynomial expansion, that the following equation is equivalent to the quadratic equation:
( x + b 2 a ) 2 = b 2 − 4 a c 4 a 2 . {\displaystyle \left(x+{\frac {b}{2a}}\right)^{2}={\frac {b^{2}-4ac}{4a^{2}}}.}
Taking the square root of both sides, and isolating x, gives:
x = − b ± b 2 − 4 a c 2 a . {\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac\ }}}{2a}}.}
Some sources, particularly older ones, use alternative parameterizations of the quadratic equation such as ax2 + 2bx + c = 0 or ax2 − 2bx + c = 0 ,[7] where b has a magnitude one half of the more common one, possibly with opposite sign. These result in slightly different forms for the solution, but are otherwise equivalent.
A number of alternative derivations can be found in the literature. These proofs are simpler than the standard completing the square method, represent interesting applications of other frequently used techniques in algebra, or offer insight into other areas of mathematics.
A lesser known quadratic formula, as used in Muller's method, and which can be found from Vieta's formulas, provides the same roots via the equation:
x = − 2 c b ± b 2 − 4 a c . {\displaystyle x={\frac {-2c}{b\pm {\sqrt {b^{2}-4ac}}}}.}
One property of this form is that it yields one valid root when a = 0, while the other root contains division by zero, because when a = 0, the quadratic equation becomes a linear equation, which has one root. By contrast, in this case the more common formula has a division by zero for one root and an indeterminate form 0/0 for the other root. On the other hand, when c = 0, the more common formula yields two correct roots whereas this form yields the zero root and an indeterminate form 0/0.
Reduced quadratic equation[edit]
It is sometimes convenient to reduce a quadratic equation so that its leading coefficient is one. This is done by dividing both sides by a, which is always possible since a is non-zero. This produces the reduced quadratic equation:[8]
x 2 + p x + q = 0 , {\displaystyle x^{2}+px+q=0,}
where p = b/a and q = c/a. This monic equation has the same solutions as the original.
The quadratic formula for the solutions of the reduced quadratic equation, written in terms of its coefficients, is:
x = 1 2 ( − p ± p 2 − 4 q ) , {\displaystyle x={\frac {1}{2}}\left(-p\pm {\sqrt {p^{2}-4q}}\right),}
or equivalently:
x = − p 2 ± ( p 2 ) 2 − q . {\displaystyle x=-{\frac {p}{2}}\pm {\sqrt {\left({\frac {p}{2}}\right)^{2}-q}}.}
Discriminant[edit]
Figure 3. Discriminant signs
In the quadratic formula, the expression underneath the square root sign is called the discriminant of the quadratic equation, and is often represented using an upper case D or an upper case Greek delta:[9]
Δ = b 2 − 4 a c . {\displaystyle \Delta =b^{2}-4ac.}
A quadratic equation with real coefficients can have either one or two distinct real roots, or two distinct complex roots. In this case the discriminant determines the number and nature of the roots. There are three cases:
If the discriminant is positive, then there are two distinct roots
− b + Δ 2 a and − b − Δ 2 a , {\displaystyle {\frac {-b+{\sqrt {\Delta }}}{2a}}\quad {\text{and}}\quad {\frac {-b-{\sqrt {\Delta }}}{2a}},}
both of which are real numbers. For quadratic equations with rational coefficients, if the discriminant is a square number, then the roots are rational—in other cases they may be quadratic irrationals.
If the discriminant is zero, then there is exactly one real root
− b 2 a , {\displaystyle -{\frac {b}{2a}},}
sometimes called a repeated or double root.
If the discriminant is negative, then there are no real roots. Rather, there are two distinct (non-real) complex roots[10]
− b 2 a + i − Δ 2 a and − b 2 a − i − Δ 2 a , {\displaystyle {\frac {-b}{2a}}+i{\frac {\sqrt {-\Delta }}{2a}}\quad {\text{and}}\quad {\frac {-b}{2a}}-i{\frac {\sqrt {-\Delta }}{2a}},}
which are complex conjugates of each other. In these expressions i is the imaginary unit.
Thus the roots are distinct if and only if the discriminant is non-zero, and the roots are real if and only if the discriminant is non-negative.
Geometric interpretation[edit]
Graph of y = ax2 + bx + c, where a and the discriminant b2 − 4ac are positive, with
Roots and y-intercept in red
Vertex and axis of symmetry in blue
Focus and directrix in pink
Visualisation of the complex roots of y = ax2 + bx + c: the parabola is rotated 180° about its vertex (orange). Its x-intercepts are rotated 90° around their mid-point, and the Cartesian plane is interpreted as the complex plane (green).[11]
The function f(x) = ax2 + bx + c is the quadratic function.[12] The graph of any quadratic function has the same general shape, which is called a parabola. The location and size of the parabola, and how it opens, depend on the values of a, b, and c. As shown in Figure 1, if a > 0, the parabola has a minimum point and opens upward. If a < 0, the parabola has a maximum point and opens downward. The extreme point of the parabola, whether minimum or maximum, corresponds to its vertex. The x-coordinate of the vertex will be located at x = − b 2 a {\displaystyle \scriptstyle x={\tfrac {-b}{2a}}} , and the y-coordinate of the vertex may be found by substituting this x-value into the function. The y-intercept is located at the point (0, c).
The solutions of the quadratic equation ax2 + bx + c = 0 correspond to the roots of the function f(x) = ax2 + bx + c, since they are the values of x for which f(x) = 0. As shown in Figure 2, if a, b, and c are real numbers and the domain of f is the set of real numbers, then the roots of f are exactly the x-coordinates of the points where the graph touches the x-axis. As shown in Figure 3, if the discriminant is positive, the graph touches the x-axis at two points; if zero, the graph touches at one point; and if negative, the graph does not touch the x-axis.
Quadratic factorization[edit]
The term
x − r {\displaystyle x-r}
is a factor of the polynomial
a x 2 + b x + c {\displaystyle ax^{2}+bx+c}
if and only if r is a root of the quadratic equation
a x 2 + b x + c = 0. {\displaystyle ax^{2}+bx+c=0.}
It follows from the quadratic formula that
a x 2 + b x + c = a ( x − − b + b 2 − 4 a c 2 a ) ( x − − b − b 2 − 4 a c 2 a ) . {\displaystyle ax^{2}+bx+c=a\left(x-{\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}\right)\left(x-{\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}}\right).}
In the special case b2 = 4ac where the quadratic has only one distinct root (i.e. the discriminant is zero), the quadratic polynomial can be factored as
a x 2 + b x + c = a ( x + b 2 a ) 2 . {\displaystyle ax^{2}+bx+c=a\left(x+{\frac {b}{2a}}\right)^{2}.}
Graphing for real roots[edit]
Figure 4. Graphing calculator computation of one of the two roots of the quadratic equation 2x2 + 4x − 4 = 0. Although the display shows only five significant figures of accuracy, the retrieved value of xc is 0.732050807569, accurate to twelve significant figures.
For most of the 20th century, graphing was rarely mentioned as a method for solving quadratic equations in high school or college algebra texts. Students learned to solve quadratic equations by factoring, completing the square, and applying the quadratic formula. Recently, graphing calculators have become common in schools and graphical methods have started to appear in textbooks, but they are generally not highly emphasized.[13]
Being able to use a graphing calculator to solve a quadratic equation requires the ability to produce a graph of y = f(x), the ability to scale the graph appropriately to the dimensions of the graphing surface, and the recognition that when f(x) = 0, x is a solution to the equation. The skills required to solve a quadratic equation on a calculator are in fact applicable to finding the real roots of any arbitrary function.
Since an arbitrary function may cross the x-axis at multiple points, graphing calculators generally require one to identify the desired root by positioning a cursor at a "guessed" value for the root. (Some graphing calculators require bracketing the root on both sides of the zero.) The calculator then proceeds, by an iterative algorithm, to refine the estimated position of the root to the limit of calculator accuracy.
Avoiding loss of significance[edit]
Although the quadratic formula provides an exact solution, the result is not exact if real numbers are approximated during the computation, as usual in numerical analysis, where real numbers are approximated by floating point numbers (called "reals" in many programming languages). In this context, the quadratic formula is not completely stable.
This occurs when the roots have different order of magnitude, or, equivalently, when b2 and b2 − 4ac are close in magnitude. In this case, the subtraction of two nearly equal numbers will cause loss of significance or catastrophic cancellation in the smaller root. To avoid this, the root that is smaller in magnitude, r, can be computed as ( c / a ) / R {\displaystyle (c/a)/R} where R is the root that is bigger in magnitude.
A second form of cancellation can occur between the terms b2 and 4ac of the discriminant, that is when the two roots are very close. This can lead to loss of up to half of correct significant figures in the roots.[7][14]
Examples and applications[edit]
The trajectory of the cliff jumper is parabolic because horizontal displacement is a linear function of time x = v x t {\displaystyle x=v_{x}t} , while vertical displacement is a quadratic function of time y = 1 2 a t 2 + v y t + h {\displaystyle y={\tfrac {1}{2}}at^{2}+v_{y}t+h} . As a result, the path follows quadratic equation y = a 2 v x 2 x 2 + v y v x x + h {\displaystyle y={\tfrac {a}{2v_{x}^{2}}}x^{2}+{\tfrac {v_{y}}{v_{x}}}x+h} , where v x {\displaystyle v_{x}} and v y {\displaystyle v_{y}} are horizontal and vertical components of the original velocity, a is gravitational acceleration and h is original height. The a value should be considered negative here, as its direction (downwards) is opposite to the height measurement (upwards).
The golden ratio is found as the solution of the quadratic equation x 2 − x − 1 = 0. {\displaystyle x^{2}-x-1=0.}
The equations of the circle and the other conic sections—ellipses, parabolas, and hyperbolas—are quadratic equations in two variables.
Given the cosine or sine of an angle, finding the cosine or sine of the angle that is half as large involves solving a quadratic equation.
The process of simplifying expressions involving the square root of an expression involving the square root of another expression involves finding the two solutions of a quadratic equation.
Descartes' theorem states that for every four kissing (mutually tangent) circles, their radii satisfy a particular quadratic equation.
The equation given by Fuss' theorem, giving the relation among the radius of a bicentric quadrilateral's inscribed circle, the radius of its circumscribed circle, and the distance between the centers of those circles, can be expressed as a quadratic equation for which the distance between the two circles' centers in terms of their radii is one of the solutions. The other solution of the same equation in terms of the relevant radii gives the distance between the circumscribed circle's center and the center of the excircle of an ex-tangential quadrilateral.
History[edit]
Babylonian mathematicians, as early as 2000 BC (displayed on Old Babylonian clay tablets) could solve problems relating the areas and sides of rectangles. There is evidence dating this algorithm as far back as the Third Dynasty of Ur.[15] In modern notation, the problems typically involved solving a pair of simultaneous equations of the form:
x + y = p , x y = q {\displaystyle x+y=p,\ \ xy=q}
which are equivalent to the equation:[16]:86
x 2 + q = p x {\displaystyle x^{2}+q=px}
The steps given by Babylonian scribes for solving the above rectangle problem were as follows:
Compute half of p.
Square the result.
Subtract q.
Find the square root using a table of squares.
Add together the results of steps (1) and (4) to give x. This is essentially equivalent to calculating x = p 2 + ( p 2 ) 2 − q {\displaystyle x={\frac {p}{2}}+{\sqrt {\left({\frac {p}{2}}\right)^{2}-q}}}
Geometric methods were used to solve quadratic equations in Babylonia, Egypt, Greece, China, and India. The Egyptian Berlin Papyrus, dating back to the Middle Kingdom (2050 BC to 1650 BC), contains the solution to a two-term quadratic equation.[17] In the Indian Sulba Sutras, circa 8th century BC, quadratic equations of the form ax2 = c and ax2 + bx = c were explored using geometric methods. Babylonian mathematicians from circa 400 BC and Chinese mathematicians from circa 200 BC used geometric methods of dissection to solve quadratic equations with positive roots.[18][19] Rules for quadratic equations were given in The Nine Chapters on the Mathematical Art, a Chinese treatise on mathematics.[19][20] These early geometric methods do not appear to have had a general formula. Euclid, the Greek mathematician, produced a more abstract geometrical method around 300 BC. With a purely geometric approach Pythagoras and Euclid created a general procedure to find solutions of the quadratic equation. In his work Arithmetica, the Greek mathematician Diophantus solved the quadratic equation, but giving only one root, even when both roots were positive.[21]
In 628 AD, Brahmagupta, an Indian mathematician, gave the first explicit (although still not completely general) solution of the quadratic equation ax2 + bx = c as follows: "To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value." (Brahmasphutasiddhanta, Colebrook translation, 1817, page 346)[16]:87 This is equivalent to:
x = 4 a c + b 2 − b 2 a . {\displaystyle x={\frac {{\sqrt {4ac+b^{2}}}-b}{2a}}.}
The Bakhshali Manuscript written in India in the 7th century AD contained an algebraic formula for solving quadratic equations, as well as quadratic indeterminate equations (originally of type ax/c = y Muhammad ibn Musa al-Khwarizmi (Persia, 9th century), inspired by Brahmagupta, developed a set of formulas that worked for positive solutions. Al-Khwarizmi goes further in providing a full solution to the general quadratic equation, accepting one or two numerical answers for every quadratic equation, while providing geometric proofs in the process.[22] He also described the method of completing the square and recognized that the discriminant must be positive,[22][23]:230 which was proven by his contemporary 'Abd al-Hamīd ibn Turk (Central Asia, 9th century) who gave geometric figures to prove that if the discriminant is negative, a quadratic equation has no solution.[23]:234 While al-Khwarizmi himself did not accept negative solutions, later Islamic mathematicians that succeeded him accepted negative solutions,[22]:191 as well as irrational numbers as solutions.[24] Abū Kāmil Shujā ibn Aslam (Egypt, 10th century) in particular was the first to accept irrational numbers (often in the form of a square root, cube root or fourth root) as solutions to quadratic equations or as coefficients in an equation.[25] The 9th century Indian mathematician Sridhara wrote down rules for solving quadratic equations.[26]
The Jewish mathematician Abraham bar Hiyya Ha-Nasi (12th century, Spain) authored the first European book to include the full solution to the general quadratic equation.[27] His solution was largely based on Al-Khwarizmi's work.[22] The writing of the Chinese mathematician Yang Hui (1238–1298 AD) is the first known one in which quadratic equations with negative coefficients of 'x' appear, although he attributes this to the earlier Liu Yi.[28] By 1545 Gerolamo Cardano compiled the works related to the quadratic equations. The quadratic formula covering all cases was first obtained by Simon Stevin in 1594.[29] In 1637 René Descartes published La Géométrie containing the quadratic formula in the form we know today. The first appearance of the general solution in the modern mathematical literature appeared in an 1896 paper by Henry Heaton.[30]
Advanced topics[edit]
Alternative methods of root calculation[edit]
Vieta's formulas[edit]
Main article: Vieta's formulas
Figure 5. Graph of the difference between Vieta's approximation for the smaller of the two roots of the quadratic equation x2 + bx + c = 0 compared with the value calculated using the quadratic formula. Vieta's approximation is inaccurate for small b but is accurate for large b. The direct evaluation using the quadratic formula is accurate for small b with roots of comparable value but experiences loss of significance errors for large b and widely spaced roots. The difference between Vieta's approximation versus the direct computation reaches a minimum at the large dots, and rounding causes squiggles in the curves beyond this minimum.
Vieta's formulas give a simple relation between the roots of a polynomial and its coefficients. In the case of the quadratic polynomial, they take the following form:
x 1 + x 2 = − b a {\displaystyle x_{1}+x_{2}=-{\frac {b}{a}}}
and
x 1 x 2 = c a . {\displaystyle x_{1}\ x_{2}={\frac {c}{a}}.}
These results follow immediately from the relation:
( x − x 1 ) ( x − x 2 ) = x 2 − ( x 1 + x 2 ) x + x 1 x 2 = 0 , {\displaystyle \left(x-x_{1}\right)\ \left(x-x_{2}\right)=x^{2}\ -\left(x_{1}+x_{2}\right)x+x_{1}x_{2}=0,}
which can be compared term by term with
x 2 + ( b / a ) x + c / a = 0. {\displaystyle x^{2}+(b/a)x+c/a=0.}
The first formula above yields a convenient expression when graphing a quadratic function. Since the graph is symmetric with respect to a vertical line through the vertex, when there are two real roots the vertex's x-coordinate is located at the average of the roots (or intercepts). Thus the x-coordinate of the vertex is given by the expression
x V = x 1 + x 2 2 = − b 2 a . {\displaystyle x_{V}={\frac {x_{1}+x_{2}}{2}}=-{\frac {b}{2a}}.}
The y-coordinate can be obtained by substituting the above result into the given quadratic equation, giving
y V = − b 2 4 a + c = − b 2 − 4 a c 4 a . {\displaystyle y_{V}=-{\frac {b^{2}}{4a}}+c=-{\frac {b^{2}-4ac}{4a}}.}
As a practical matter, Vieta's formulas provide a useful method for finding the roots of a quadratic in the case where one root is much smaller than the other. If | x 2| << | x 1|, then x 1 + x 2 ≈ x 1, and we have the estimate:
x 1 ≈ − b a . {\displaystyle x_{1}\approx -{\frac {b}{a}}.}
The second Vieta's formula then provides:
x 2 = c a x 1 ≈ − c b . {\displaystyle x_{2}={\frac {c}{a\ x_{1}}}\approx -{\frac {c}{b}}.}
These formulas are much easier to evaluate than the quadratic formula under the condition of one large and one small root, because the quadratic formula evaluates the small root as the difference of two very nearly equal numbers (the case of large b), which causes round-off error in a numerical evaluation. Figure 5 shows the difference between (i) a direct evaluation using the quadratic formula (accurate when the roots are near each other in value) and (ii) an evaluation based upon the above approximation of Vieta's formulas (accurate when the roots are widely spaced). As the linear coefficient b increases, initially the quadratic formula is accurate, and the approximate formula improves in accuracy, leading to a smaller difference between the methods as b increases. However, at some point the quadratic formula begins to lose accuracy because of round off error, while the approximate method continues to improve. Consequently, the difference between the methods begins to increase as the quadratic formula becomes worse and worse.
This situation arises commonly in amplifier design, where widely separated roots are desired to ensure a stable operation (see step response).
Trigonometric solution[edit]
In the days before calculators, people would use mathematical tables—lists of numbers showing the results of calculation with varying arguments—to simplify and speed up computation. Tables of logarithms and trigonometric functions were common in math and science textbooks. Specialized tables were published for applications such as astronomy, celestial navigation and statistics. Methods of numerical approximation existed, called prosthaphaeresis, that offered shortcuts around time-consuming operations such as multiplication and taking powers and roots.[13] Astronomers, especially, were concerned with methods that could speed up the long series of computations involved in celestial mechanics calculations.
It is within this context that we may understand the development of means of solving quadratic equations by the aid of trigonometric substitution. Consider the following alternate form of the quadratic equation,
[1] a x 2 + b x ± c = 0 , {\displaystyle ax^{2}+bx\pm c=0,}
where the sign of the ± symbol is chosen so that a and c may both be positive. By substituting
[2] x = c / a tan  θ {\displaystyle x={\sqrt {c/a}}\tan \theta }
and then multiplying through by cos2θ, we obtain
[3] sin 2  θ + b a c sin  θ cos  θ ± cos 2  θ = 0. {\displaystyle \sin ^{2}\theta +{\frac {b}{\sqrt {ac}}}\sin \theta \cos \theta \pm \cos ^{2}\theta =0.}
Introducing functions of 2θ and rearranging, we obtain
[4] tan  2 θ n = + 2 a c b , {\displaystyle \tan 2\theta _{n}=+2{\frac {\sqrt {ac}}{b}},}
[5] sin  2 θ p = − 2 a c b , {\displaystyle \sin 2\theta _{p}=-2{\frac {\sqrt {ac}}{b}},}
where the subscripts n and p correspond, respectively, to the use of a negative or positive sign in equation [1]. Substituting the two values of θn or θp found from equations [4] or [5] into [2] gives the required roots of [1]. Complex roots occur in the solution based on equation [5] if the absolute value of sin 2θp exceeds unity. The amount of effort involved in solving quadratic equations using this mixed trigonometric and logarithmic table look-up strategy was two-thirds the effort using logarithmic tables alone.[31] Calculating complex roots would require using a different trigonometric form.[32]
To illustrate, let us assume we had available seven-place logarithm and trigonometric tables, and wished to solve the following to six-significant-figure accuracy:
4.16130 x 2 + 9.15933 x − 11.4207 = 0 {\displaystyle 4.16130x^{2}+9.15933x-11.4207=0}
A seven-place lookup table might have only 100,000 entries, and computing intermediate results to seven places would generally require interpolation between adjacent entries.
log  a = 0.6192290 , log  b = 0.9618637 , log  c = 1.0576927 {\displaystyle \log a=0.6192290,\log b=0.9618637,\log c=1.0576927}
2 a c / b = 2 × 10 ( 0.6192290 + 1.0576927 ) / 2 − 0.9618637 = 1.505314 {\displaystyle 2{\sqrt {ac}}/b=2\times 10^{(0.6192290+1.0576927)/2-0.9618637}=1.505314}
θ = ( tan − 1  1.505314 ) / 2 = 28.20169 ∘ or − 61.79831 ∘ {\displaystyle \theta =(\tan ^{-1}1.505314)/2=28.20169^{\circ }{\text{ or }}-61.79831^{\circ }}
log  | tan  θ | = − 0.2706462 or 0.2706462 {\displaystyle \log |\tan \theta |=-0.2706462{\text{ or }}0.2706462}
log  c / a = ( 1.0576927 − 0.6192290 ) / 2 = 0.2192318 {\displaystyle \log {\sqrt {c/a}}=(1.0576927-0.6192290)/2=0.2192318}
x 1 = 10 0.2192318 − 0.2706462 = 0.888353 {\displaystyle x_{1}=10^{0.2192318-0.2706462}=0.888353} (rounded to six significant figures)
x 2 = − 10 0.2192318 + 0.2706462 = − 3.08943 {\displaystyle x_{2}=-10^{0.2192318+0.2706462}=-3.08943}
Solution for complex roots in polar coordinates[edit]
If the quadratic equation a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0} with real coefficients has two complex roots—the case where b 2 − 4 a c < 0 , {\displaystyle b^{2}-4ac<0,} requiring a and c to have the same sign as each other—then the solutions for the roots can be expressed in polar form as[33]
x 1 , x 2 = r ( cos  θ ± i sin  θ ) , {\displaystyle x_{1},\,x_{2}=r(\cos \theta \pm i\sin \theta ),}
where r = c a {\displaystyle r={\sqrt {\tfrac {c}{a}}}} and θ = cos − 1  ( − b 2 a c ) . {\displaystyle \theta =\cos ^{-1}\left({\tfrac {-b}{2{\sqrt {ac}}}}\right).}
Geometric solution[edit]
Figure 6. Geometric solution of ax2 + bx + c = 0 using Lill's method. Solutions are −AX1/SA, −AX2/SA
The quadratic equation may be solved geometrically in a number of ways. One way is via Lill's method. The three coefficients a, b, c are drawn with right angles between them as in SA, AB, and BC in Figure 6. A circle is drawn with the start and end point SC as a diameter. If this cuts the middle line AB of the three then the equation has a solution, and the solutions are given by negative of the distance along this line from A divided by the first coefficient a or SA. If a is 1 the coefficients may be read off directly. Thus the solutions in the diagram are −AX1/SA and −AX2/SA.[34]
Carlyle circle of the quadratic equation x2 − sx + p = 0.
The Carlyle circle, named after Thomas Carlyle, has the property that the solutions of the quadratic equation are the horizontal coordinates of the intersections of the circle with the horizontal axis.[35] Carlyle circles have been used to develop ruler-and-compass constructions of regular polygons.
Generalization of quadratic equation[edit]
The formula and its derivation remain correct if the coefficients a, b and c are complex numbers, or more generally members of any field whose characteristic is not 2. (In a field of characteristic 2, the element 2a is zero and it is impossible to divide by it.)
The symbol
± b 2 − 4 a c {\displaystyle \pm {\sqrt {b^{2}-4ac}}}
in the formula should be understood as "either of the two elements whose square is b2 − 4ac, if such elements exist". In some fields, some elements have no square roots and some have two; only zero has just one square root, except in fields of characteristic 2. Even if a field does not contain a square root of some number, there is always a quadratic extension field which does, so the quadratic formula will always make sense as a formula in that extension field.
Characteristic 2[edit]
In a field of characteristic 2, the quadratic formula, which relies on 2 being a unit, does not hold. Consider the monic quadratic polynomial
x 2 + b x + c {\displaystyle x^{2}+bx+c}
over a field of characteristic 2. If b = 0, then the solution reduces to extracting a square root, so the solution is
x = c {\displaystyle x={\sqrt {c}}}
and there is only one root since
− c = − c + 2 c = c . {\displaystyle -{\sqrt {c}}=-{\sqrt {c}}+2{\sqrt {c}}={\sqrt {c}}.}
In summary,
x 2 + c = ( x + c ) 2 . {\displaystyle \displaystyle x^{2}+c=(x+{\sqrt {c}})^{2}.}
See quadratic residue for more information about extracting square roots in finite fields.
In the case that b ≠ 0, there are two distinct roots, but if the polynomial is irreducible, they cannot be expressed in terms of square roots of numbers in the coefficient field. Instead, define the 2-root R(c) of c to be a root of the polynomial x2 + x + c, an element of the splitting field of that polynomial. One verifies that R(c) + 1 is also a root. In terms of the 2-root operation, the two roots of the (non-monic) quadratic ax2 + bx + c are
b a R ( a c b 2 ) {\displaystyle {\frac {b}{a}}R\left({\frac {ac}{b^{2}}}\right)}
and
b a ( R ( a c b 2 ) + 1 ) . {\displaystyle {\frac {b}{a}}\left(R\left({\frac {ac}{b^{2}}}\right)+1\right).}
For example, let a denote a multiplicative generator of the group of units of F4, the Galois field of order four (thus a and a + 1 are roots of x2 + x + 1 over F4. Because (a + 1)2 = a, a + 1 is the unique solution of the quadratic equation x2 + a = 0. On the other hand, the polynomial x2 + ax + 1 is irreducible over F4, but it splits over F16, where it has the two roots ab and ab + a, where b is a root of x2 + x + a in F16.
This is a special case of Artin–Schreier theory.
See also[edit]
Chakravala method
Completing the square
Cubic function
Fundamental theorem of algebra
Linear equation
Parabola
Periodic points of complex quadratic mappings
Quadratic function
Quadratic polynomial
Quartic function
Quintic function
Solving quadratic equations with continued fractions
References[edit]
Jump up ^ Protters & Morrey: " Calculus and Analytic Geometry. First Course"
^ Jump up to: a b c Washington, Allyn J. (2000). Basic Technical Mathematics with Calculus, Seventh Edition. Addison Wesley Longman, Inc. ISBN 0-201-35666-X.
Jump up ^ Ebbinghaus, Heinz-Dieter; Ewing, John H. (1991), Numbers, Graduate Texts in Mathematics, 123, Springer, p. 77, ISBN 9780387974972 .
Jump up ^ Sterling, Mary Jane (2010), Algebra I For Dummies, Wiley Publishing, p. 219, ISBN 978-0-470-55964-2
Jump up ^ Rich, Barnett; Schmidt, Philip (2004), Schaum's Outline of Theory and Problems of Elementary Algebra, The McGraw-Hill Companies, ISBN 0-07-141083-X , Chapter 13 §4.4, p. 291
Jump up ^ Himonas, Alex. Calculus for Business and Social Sciences, p. 64 (Richard Dennis Publications, 2001).
^ Jump up to: a b Kahan, Willian (November 20, 2004), On the Cost of Floating-Point Computation Without Extra-Precise Arithmetic (PDF), retrieved 2012-12-25
Jump up ^ Alenit͡syn, Aleksandr and Butikov, Evgeniĭ. Concise Handbook of Mathematics and Physics, p. 38 (CRC Press 1997)
Jump up ^ Δ is the initial of the Greek word Διακρίνουσα, Diakrínousa, discriminant.
Jump up ^ Achatz, Thomas; Anderson, John G.; McKenzie, Kathleen (2005). Technical Shop Mathematics. Industrial Press. p. 277. ISBN 0-8311-3086-5.
Jump up ^ "Complex Roots Made Visible – Math Fun Facts". Retrieved 1 October 2016.
Jump up ^ Wharton, P. (2006). Essentials of Edexcel Gcse Math/Higher. Lonsdale. p. 63. ISBN 978-1-905-129-78-2.
^ Jump up to: a b Ballew, Pat. "Solving Quadratic Equations — By analytic and graphic methods; Including several methods you may never have seen" (PDF). Retrieved 18 April 2013.
Jump up ^ Higham, Nicholas (2002), Accuracy and Stability of Numerical Algorithms (2nd ed.), SIAM, p. 10, ISBN 978-0-89871-521-7
Jump up ^ Friberg, Jöran (2009). "A Geometric Algorithm with Solutions to Quadratic Equations in a Sumerian Juridical Document from Ur III Umma". Cuneiform Digital Library Journal. 3.
^ Jump up to: a b Stillwell, John (2004). Mathematics and Its History (2nd ed.). Springer. ISBN 0-387-95336-1.
Jump up ^ The Cambridge Ancient History Part 2 Early History of the Middle East. Cambridge University Press. 1971. p. 530. ISBN 978-0-521-07791-0.
Jump up ^ Henderson, David W. "Geometric Solutions of Quadratic and Cubic Equations". Mathematics Department, Cornell University. Retrieved 28 April 2013.
^ Jump up to: a b Aitken, Wayne. "A Chinese Classic: The Nine Chapters" (PDF). Mathematics Department, California State University. Retrieved 28 April 2013.
Jump up ^ Smith, David Eugene (1958). History of Mathematics. Courier Dover Publications. p. 380. ISBN 978-0-486-20430-7.
Jump up ^ Smith, David Eugene (1958). History of Mathematics, Volume 1. Courier Dover Publications. p. 134. ISBN 0-486-20429-4. Extract of page 134
^ Jump up to: a b c d Katz, V. J.; Barton, B. (2006). "Stages in the History of Algebra with Implications for Teaching". Educational Studies in Mathematics. 66 (2): 185–201. doi:10.1007/s10649-006-9023-7.
^ Jump up to: a b Boyer, Carl B.; Uta C. Merzbach, rev. editor (1991). A History of Mathematics. John Wiley & Sons, Inc. ISBN 0-471-54397-7. CS1 maint: Extra text: authors list (link)
Jump up ^ O'Connor, John J.; Robertson, Edmund F., "Arabic mathematics: forgotten brilliance?", MacTutor History of Mathematics archive, University of St Andrews . "Algebra was a unifying theory which allowed rational numbers, irrational numbers, geometrical magnitudes, etc., to all be treated as "algebraic objects"."
Jump up ^ Jacques Sesiano, "Islamic mathematics", p. 148, in Selin, Helaine; D'Ambrosio, Ubiratan, eds. (2000), Mathematics Across Cultures: The History of Non-Western Mathematics, Springer, ISBN 1-4020-0260-2
Jump up ^ Smith, David Eugene (1958). History of Mathematics. Courier Dover Publications. p. 280. ISBN 978-0-486-20429-1.
Jump up ^ Livio, Mario (2006). The Equation that Couldn't Be Solved. Simon & Schuster. ISBN 0743258215.
Jump up ^ Ronan, Colin (1985). The Shorter Science and Civilisation in China. Cambridge University Press. p. 15. ISBN 978-0-521-31536-4.
Jump up ^ Struik, D. J.; Stevin, Simon (1958), The Principal Works of Simon Stevin, Mathematics (PDF), II–B, C. V. Swets & Zeitlinger, p. 470
Jump up ^ Heaton, H (1896). "A Method of Solving Quadratic Equations". American Mathematical Monthly. 3 (10): 236–237. JSTOR 2971099. doi:10.2307/2971099.
Jump up ^ Seares, F. H. (1945). "Trigonometric Solution of the Quadratic Equation". Publications of the Astronomical Society of the Pacific. 57 (339): 307–309. Bibcode:1945PASP...57..307S. doi:10.1086/125759.
Jump up ^ Aude, H. T. R. (1938). "The Solutions of the Quadratic Equation Obtained by the Aid of the Trigonometry". National Mathematics Magazine. 13 (3): 118–121. JSTOR 3028750. doi:10.2307/3028750.
Jump up ^ Simons, Stuart, "Alternative approach to complex roots of real quadratic equations", Mathematical Gazette 93, March 2009, 91–92.
Jump up ^ Bixby, William Herbert (1879), Graphical Method for finding readily the Real Roots of Numerical Equations of Any Degree, West Point N. Y.
Jump up ^ Weisstein, Eric W. "Carlyle Circle". From MathWorld—A Wolfram Web Resource. Retrieved 21 May 2013.
External links[edit]
Wikimedia Commons has media related to Quadratic equation.
Hazewinkel, Michiel, ed. (2001) [1994], "Quadratic equation", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Weisstein, Eric W. "Quadratic equations". MathWorld.
101 uses of a quadratic equation
101 uses of a quadratic equation: Part II
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Polynomial equation

values: Edit section: Solving the quadratic equation

values: Figure 1. Plots of the quadratic function, y = eh x squared plus b x plus c, varying each coefficient separately while the other coefficients are fixed at values eh = 1, b = 0, c = 0. The left plot illustrates varying c. When c equals 0, the vertex of the parabola representing the quadratic function is centered on the origin, and the parabola rises on both sides of the origin, opening to the top. When c is greater than zero, the parabola does not change in shape, but its vertex is raised above the origin. When c is less than zero, the vertex of the parabola is lowered below the origin. The center plot illustrates varying b. When b is less than zero, the parabola representing the quadratic function is unchanged in shape, but its vertex is shifted to the right of and below the origin. When b is greater than zero, its vertex is shifted to the left of and below the origin. The vertices of the family of curves created by varying b follow along a parabolic curve. The right plot illustrates varying eh. When eh is positive, the quadratic function is a parabola opening to the top. When eh is zero, the quadratic function is a horizontal straight line. When eh is negative, the quadratic function is a parabola opening to the bottom.

values: Figure 2 illustrates an x y plot of the quadratic function f of x equals x squared minus x minus 2. The x-coordinate of the points where the graph intersects the x-axis, x equals −1 and x equals 2, are the solutions of the quadratic equation x squared minus x minus 2 equals zero.

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values: Edit section: Quadratic formula and its derivation

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values: Edit section: Reduced quadratic equation

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Leading coefficient

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values: Figure 3. This figure plots three quadratic functions on a single Cartesian plane graph to illustrate the effects of discriminant values. When the discriminant, delta, is positive, the parabola intersects the x-axis at two points. When delta is zero, the vertex of the parabola touches the x-axis at a single point. When delta is negative, the parabola does not intersect the x-axis at all.

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values: \frac{-b + \sqrt {\Delta}}{2a} \quad\text{and}\quad \frac{-b - \sqrt {\Delta}}{2a},

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Quadratic irrational

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values: \frac{-b}{2a} + i \frac{\sqrt {-\Delta}}{2a} \quad\text{and}\quad \frac{-b}{2a} - i \frac{\sqrt {-\Delta}}{2a},

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values: ax^2+bx+c = a \left( x - \frac{-b + \sqrt {b^2-4ac}}{2a} \right) \left( x - \frac{-b - \sqrt {b^2-4ac}}{2a} \right).

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values: ax^2+bx+c = a \left( x + \frac{b}{2a} \right)^2.

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Graphing calculators

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Catastrophic cancellation

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values: x = \frac{p}{2} + \sqrt{\left(\frac{p}{2}\right)^2 - q}

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values: Edit section: Alternative methods of root calculation

values: Figure 5. Graph of the difference between Vieta's approximation for the smaller of the two roots of the quadratic equation x squared plus b x plus c equals zero compared with the value calculated using the quadratic formula. The difference is plotted as a function of b for two different values of c, c equals 4, and c equals 400,000. The graph is a log log graph, with the vertical axis, the difference, ranging from ten to the minus 13 at the bottom to ten to the minus 1 at the top. The horizontal axis, b, ranges from 10 at the left to ten to the eighth at the right. Vieta's approximation for the smaller root is not accurate for small b but is accurate for large b. The direct evaluation of the smaller root using the quadratic formula is accurate for small b with roots of comparable value, but experiences loss of significance errors for large b and widely spaced roots. When c equals 4, Vieta's approximation starts off poorly at the left, but gets better with larger b, the difference between Vieta's approximation and the quadratic formula reaching a minimum at approximately b equals ten to the fifth. Vieta's approximation and the quadratic formula then start diverging again because the quadratic formula experiences loss of significance error. When c equals four hundred thousand, the difference between Vieta's approximation and the quadratic formula reaches a minimum at approximately b equals ten to the seventh. The curves are both straight to the left of the minimum, indicating a simple monomial power relationship between the difference and b. Likewise, the curves are both approximately straight to the right of the minimum, indicating a power relationship, except that the straight lines have squiggles in them due to the loss of significance errors in the quadratic formula.

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values: \left( x - x_1 \right) \ \left( x-x_2 \right ) = x^2 \ - \left( x_1+x_2 \right)x +x_1 x_2 = 0,

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values: x_2 = \frac{c}{a \ x_1} \approx -\frac{c}{b} .

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values: \sin^2\theta + \frac{b}{\sqrt {ac}} \sin\theta  \cos\theta \pm \cos^2\theta = 0 .

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values: \tan 2 \theta_n = + 2 \frac{\sqrt{ac}}{b} ,

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values: \sin 2 \theta_p = - 2 \frac{\sqrt{ac}}{b} ,

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values: Edit section: Solution for complex roots in polar coordinates

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values: x_{1},\,x_{2}=r(\cos \theta \pm i\sin \theta ),

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values: \theta =\cos ^{{-1}}\left({\tfrac  {-b}{2{\sqrt  {ac}}}}\right).

values: Figure 6. Geometric solution of eh x squared plus b x plus c = 0 using Lill's method. The geometric construction is as follows: Draw a trapezoid S Eh B C. Line S Eh of length eh is the vertical left side of the trapezoid. Line Eh B of length b is the horizontal bottom of the trapezoid. Line B C of length c is the vertical right side of the trapezoid. Line C S completes the trapezoid. From the midpoint of line C S, draw a circle passing through points C and S. Depending on the relative lengths of eh, b, and c, the circle may or may not intersect line Eh B. If it does, then the equation has a solution. If we call the intersection points X 1 and X 2, then the two solutions are given by negative Eh X 1 divided by S Eh, and negative Eh X 2 divided by S Eh.

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values: -\sqrt{c} = -\sqrt{c} + 2\sqrt{c} = \sqrt{c}.

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values: \displaystyle x^{2} + c = (x + \sqrt{c})^{2}.

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Passive immunity
From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
Passive immunity is the transfer of active humoral immunity in the form of ready-made antibodies. Passive immunity can occur naturally, when maternal antibodies are transferred to the fetus through the placenta, and it can also be induced artificially, when high levels of antibodies specific to a pathogen or toxin (obtained from humans, horses, or other animals) are transferred to non-immune persons through blood products that contain antibodies, such as in immunoglobulin therapy or antiserum therapy.[1] Passive immunization is used when there is a high risk of infection and insufficient time for the body to develop its own immune response, or to reduce the symptoms of ongoing or immunosuppressive diseases.[2] Passive immunization can be provided when people cannot synthesize antibodies, and when they have been exposed to a disease that they do not have immunity against.[3]
Contents
[hide]
1 Naturally acquired
2 Artificially acquired
2.1 History and applications of artificial passive immunity
2.2 FDA licensed immunoglobulins
3 Passive transfer of cell-mediated immunity
4 Advantages and disadvantages
5 See also
6 References
Naturally acquired[edit]
Maternal passive immunity is a type of naturally acquired passive immunity, and refers to antibody-mediated immunity conveyed to a fetus or infant by its mother. Naturally acquired passive immunity can be provided during pregnancy, and through breastfeeding.[4] In humans, maternal antibodies (MatAb) are passed through the placenta to the fetus by an FcRn receptor on placental cells. This occurs predominately during the third trimester of pregnancy, and thus is often reduced in babies born prematurely. Immunoglobulin G (IgG) is the only antibody isotype that can pass through the human placenta, and is the most common antibody of the five types of antibodies found in the body. IgG antibodies protects against bacterial and viral infections in fetuses. Immunization is often required shortly following birth to prevent diseases in newborns such as tuberculosis, hepatitis B, polio, and pertussis, however, maternal IgG can inhibit the induction of protective vaccine responses throughout the first year of life. This effect is usually overcome by secondary responses to booster immunization.[5]
Passive immunity is also provided through colostrum and breast milk, which contain IgA antibodies that are transferred to the gut of the infant, providing local protection against disease causing bacteria and viruses until the newborn can synthesize its own antibodies.[6] Maternal antibodies protect against some diseases more than others such as measles, rubella, and tetanus compared to the protection provided against polio, and pertussis.[7] Maternal passive immunity offers immediate protection, though protection mediated by maternal IgG typically only lasting four to six months after birth.[8] Protection mediated by IgA is dependent on the length of time that an infant is breastfed, which is one of the reasons the World Health Organization recommends breastfeeding for at least the first two years of life.[9]
Only a few other species besides humans transfer maternal antibodies before birth, including primates and lagomorphs (which includes rabbits and hares).[10] In some of these species IgM can be transferred across the placenta as well as IgG. All other mammalian species predominantly or solely transfer maternal antibodies after birth through milk. In these species, the neonatal gut is able to absorb IgG for hours to days after birth. However, after a period of time the neonate can no longer absorb maternal IgG through their gut, an event that is referred to as "gut closure". If a neonatal animal does not receive adequate amounts of colostrum prior to gut closure, is does not have a sufficient amount of maternal IgG in its blood to fight off common diseases. This condition is referred to as failure of passive transfer. It can be diagnosed by measuring the amount of IgG in a newborn's blood, and is treated with intravenous administration of immunoglobulins. If not treated, it can be fatal.
Artificially acquired[edit]
See also: Temporarily induced immunity
Artificially acquired passive immunity is a short-term immunization achieved by the transfer of antibodies, which can be administered in several forms; as human or animal blood plasma or serum, as pooled human immunoglobulin for intravenous (IVIG) or intramuscular (IG) use, as high-titer human IVIG or IG from immunized donors or from donors recovering from the disease, and as monoclonal antibodies (MAb). Passive transfer is used to prevent disease or used prophylactically in the case of immunodeficiency diseases, such as hypogammaglobulinemia.[11][12] It is also used in the treatment of several types of acute infection, and to treat poisoning.[2] Immunity derived from passive immunization lasts for a few weeks to three to four months.[13][14] There is also a potential risk for hypersensitivity reactions, and serum sickness, especially from gamma globulin of non-human origin.[6] Passive immunity provides immediate protection, but the body does not develop memory, therefore the patient is at risk of being infected by the same pathogen later unless they acquire active immunity or vaccination.[6]
History and applications of artificial passive immunity[edit]
A vial of diphtheria antitoxin, dated 1895
In 1888 Emile Roux and Alexandre Yersin showed that the clinical effects of diphtheria were caused by diphtheria toxin and, following the 1890 discovery of an antitoxin-based immunity to diphtheria and tetanus by Emil Adolf von Behring and Kitasato Shibasaburō, antitoxin became the first major success of modern therapeutic immunology.[15][16] Shibasaburo and von Behring immunized guinea pigs with the blood products from animals that had recovered from diphtheria and realized that the same process of heat treating blood products of other animals could treat humans with diphtheria.[17] By 1896, the introduction of diphtheria antitoxin was hailed as "the most important advance of the [19th] Century in the medical treatment of acute infective disease".[18]
Prior to the advent of vaccines and antibiotics, specific antitoxin was often the only treatment available for infections such as diphtheria and tetanus. Immunoglobulin therapy continued to be a first line therapy in the treatment of severe respiratory diseases until the 1930s, even after sulfonamides were introduced.[12]
This image is from the Historical Medical Library of The College of Physicians of Philadelphia. This displays the administration of diphtheria antitoxin from horse serum to young child, dated 1895.
In 1890 antibody therapy was used to treat tetanus, when serum from immunized horses was injected into patients with severe tetanus in an attempt to neutralize the tetanus toxin, and prevent the dissemination of the disease. Since the 1960s, human tetanus immune globulin (TIG) has been used in the United States in unimmunized, vaccine-naive or incompletely immunized patients who have sustained wounds consistent with the development of tetanus.[12] The administration of horse antitoxin remains the only specific pharmacologic treatment available for botulism.[19] Antitoxin also known as heterologous hyperimmune serum is often also given prophylactically to individuals known to have ingested contaminated food.[20] IVIG treatment was also used successfully to treat several victims of toxic shock syndrome, during the 1970s tampon scare.
Antibody therapy is also used to treat viral infections. In 1945, hepatitis A infections, epidemic in summer camps, were successfully prevented by immunoglobulin treatment. Similarly, hepatitis B immune globulin (HBIG) effectively prevents hepatitis B infection. Antibody prophylaxis of both hepatitis A and B has largely been supplanted by the introduction of vaccines; however, it is still indicated following exposure and prior to travel to areas of endemic infection.[21]
In 1953, human vaccinia immunoglobulin (VIG) was used to prevent the spread of smallpox during an outbreak in Madras, India, and continues to be used to treat complications arising from smallpox vaccination. Although the prevention of measles is typically induced through vaccination, it is often treated immuno-prophylactically upon exposure. Prevention of rabies infection still requires the use of both vaccine and immunoglobulin treatments.[12]
During a 1995 Ebola virus outbreak in the Democratic Republic of Congo, whole blood from recovering patients, and containing anti-Ebola antibodies, was used to treat eight patients, as there was no effective means of prevention, though a treatment was discovered recently in the 2013 Ebola epidemic in Africa. Only one of the eight infected patients died, compared to a typical 80% Ebola mortality, which suggested that antibody treatment may contribute to survival.[22] Immune globulin or immunoglobulin has been used to both prevent and treat reactivation of the herpes simplex virus (HSV), varicella zoster virus, Epstein-Barr virus (EBV), and cytomegalovirus (CMV).[12]
FDA licensed immunoglobulins[edit]
The following immunoglobulins are the immunoglubulins currently approved for use for infectious disease prophylaxis and immunotherapy, in the United States.[23]
FDA approved products for passive immunization and immunotherapy
Disease
Product
Source
Use
Botulism Specific equine IgG horse Treatment of wound and food borne forms of botulism, infant
botulism is treated with human botulism immune globulin (BabyBIG).
Cytomegalovirus (CMV) hyper-immune IVIG human Prophylaxis, used most often in kidney transplant patients.
Diphtheria Specific equine IgG horse Treatment of diphtheria infection.
Hepatitis A, measles Pooled human Ig human serum Prevention of Hepatitis A and measles infection,
treatment of congenital or acquired immunodeficiency.
Hepatitis B Hepatitis B Ig human Post-exposure prophylaxis, prevention in high-risk infants
(administered with Hepatitis B vaccine).
ITP, Kawasaki disease,
IgG deficiency Pooled human IgG human serum Treatment of ITP and Kawasaki disease,
prevention/treatment of opportunistic infection with IgG deficiency.
Rabies Rabies Ig human Post-exposure prophylaxis (administered with rabies vaccine).
Tetanus Tetanus Ig human Treatment of tetanus infection.
Vaccinia Vaccinia Ig human Treatment of progressive vaccinia infection
including eczema and ocular forms (usually resulting from
smallpox vaccination in immunocompromised individuals).
Varicella (chicken-pox) Varicella-zoster Ig human Post-exposure prophylaxis in high risk individuals.
Passive transfer of cell-mediated immunity[edit]
The one exception to passive humoral immunity is the passive transfer of cell-mediated immunity, also called adoptive immunization which involves the transfer of mature circulating lymphocytes. It is rarely used in humans, and requires histocompatible (matched) donors, which are often difficult to find, and carries severe risks of graft-versus-host disease.[2] This technique has been used in humans to treat certain diseases including some types of cancer and immunodeficiency. However, this specialized form of passive immunity is most often used in a laboratory setting in the field of immunology, to transfer immunity between "congenic", or deliberately inbred mouse strains which are histocompatible.
Advantages and disadvantages[edit]
An individual's immune response of passive immunity is "faster than a vaccine" and can instill immunity in an individual that does not "respond to immunization", often within hours or a few days. Another advantage to passive immunity is that maternal antibodies that are passed to the newborn during nursing boosts the newborn's immune system having lasting effects on the baby's health such as decreased risk of obesity.[17][24]
A disadvantage to passive immunity is that producing antibodies in a laboratory is expensive and difficult to do. In order to produce antibodies for infectious diseases, there is a need for possibly thousands of human donors to donate blood or immune animals' blood would be obtained for the antibodies. Patients who are immunized with the antibodies from animals may develop serum sickness due to the proteins from the immune animal and develop serious allergic reactions.[25] Antibody treatments can be time consuming and are given through an intravenous injection or IV, while a vaccine shot or jab is less time consuming and has less risk of complication than an antibody treatment. Passive immunity is effective, but ephemeral (lasting a short amount of time).[17]
See also[edit]
Active immunity
Immunity (medical)
References[edit]
Jump up ^ "Vaccines: Vac-Gen/Immunity Types". www.cdc.gov. Retrieved 2015-11-20.
^ Jump up to: a b c Microbiology and Immunology On-Line Textbook: USC School of Medicine
Jump up ^ "Passive Immunization - Infectious Diseases". Merck Manuals Professional Edition. Retrieved 2015-11-12.
Jump up ^ Kalenik, Barbara; Sawicka, Róża; Góra-Sochacka, Anna; Sirko, Agnieszka (2014-01-01). "Influenza prevention and treatment by passive immunization". Acta Biochimica Polonica. 61 (3): 573–587. ISSN 1734-154X. PMID 25210721.
Jump up ^ Lambert, Paul-Henri, Margaret Liu and Claire-Anne Siegrist Can successful vaccines teach us how to induce efficient protective immune responses? (Full text-html) Nature Medicine 11, S54 - S62 (2005).
^ Jump up to: a b c Janeway, Charles; Paul Travers; Mark Walport; Mark Shlomchik (2001). Immunobiology; Fifth Edition. New York and London: Garland Science. ISBN 0-8153-4101-6. .
Jump up ^ "Centers for Disease Control and Prevention" (PDF).
Jump up ^ "Natural Passive Immunity - Boundless Open Textbook". Boundless. Retrieved 2015-11-12.
Jump up ^ "WHO | Exclusive breastfeeding". www.who.int. Retrieved 2016-06-06.
Jump up ^ Mucosal Immunology. ISBN 9780124158474.
Jump up ^ "prophylactically".
^ Jump up to: a b c d e Keller, Margaret A. and E. Richard Stiehm Passive Immunity in Prevention and Treatment of Infectious Diseases Clinical Microbiology Reviews, October 2000, p. 602-614, Vol. 13, No. 4
Jump up ^ Immunity Types from Centers for Disease Control and Prevention. Page last updated: May 19, 2014
Jump up ^ Baxter, David (2007-12-01). "Active and passive immunity, vaccine types, excipients and licensing". Occupational Medicine. 57 (8): 552–556. doi:10.1093/occmed/kqm110. ISSN 0962-7480. PMID 18045976.
Jump up ^ Dolman, C.E. (1973). "Landmarks and pioneers in the control of diphtheria". Can. J. Public Health. 64 (4): 317–36. PMID 4581249.
Jump up ^ Silverstein, Arthur M. (1989) History of Immunology (Hardcover) Academic Press. Note: The first six pages of this text are available online at: (Amazon.com easy reader)
^ Jump up to: a b c "Passive Immunization — History of Vaccines". www.historyofvaccines.org. Retrieved 2015-11-20.
Jump up ^ (Report) (1896). "Report of the Lancet Special Commission on the relative strengths of diphtheria antitoxic serums". Lancet. 148 (3803): 182–95. doi:10.1016/s0140-6736(01)72399-9.
Jump up ^ Shapiro, Roger L. MD; Charles Hatheway, PhD; and David L. Swerdlow, MD Botulism in the United States: A Clinical and Epidemiologic Review Annals of Internal Medicine. 1 August 1998 Volume 129 Issue 3 Pages 221-228
Jump up ^ "Centers of Disease Control and Prevention" (PDF).
Jump up ^ Casadevall, A., and M. D. Scharff. 1995. Return to the past: the case for antibody-based therapies in infectious diseases. Clin. Infect. Dis. 21:150-161
Jump up ^ Mupapa, K., M. Massamba, K. Kibadi, K. Kivula, A. Bwaka, M. Kipasa, R. Colebunders, and J. J. Muyembe-Tamfum on behalf of the International Scientific and Technical Committee. 1999. Treatment of Ebola hemorrhagic fever with blood transfusions from convalescent patients. J. Infect. Dis. 179(Suppl.):S18-S23
Jump up ^ Samuel Baron MD (1996) Table 8-2. U.S. Licensed Immunoglobulin For Passive Immunization Medical Microbiology Fourth Edition The University of Texas Medical Branch at Galveston
Jump up ^ "Breastfeeding Overview". WebMD. Retrieved 2015-11-20.
Jump up ^ "Centers for Disease Control and Prevention" (PDF).
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