Quadratic equation

In mathematics, a quadratic equation (Template:Etymology) is an equation that can be rearranged in standard form as<ref>Template:Cite book Extract of page 219</ref> <math display=block>ax^2 + bx + c = 0\,,</math> where the variable Template:Math represents an unknown number, and Template:Math, Template:Math, and Template:Math represent known numbers, where Template:Math. (If Template:Math and Template:Math then the equation is linear, not quadratic.) The numbers Template:Math, Template:Math, and Template:Math are the coefficients of the equation and may be distinguished by respectively calling them, the quadratic coefficient, the linear coefficient and the constant coefficient or free term.<ref>Protters & Morrey: "Calculus and Analytic Geometry. First Course".</ref>

The values of Template:Mvar that satisfy the equation are called solutions of the equation, and roots or zeros of the quadratic function on its left-hand side. A quadratic equation has at most two solutions. If there is only one solution, one says that it is a double root. If all the coefficients are real numbers, there are either two real solutions, or a single real double root, or two complex solutions that are complex conjugates of each other. A quadratic equation always has two roots, if complex roots are included and a double root is counted for two. A quadratic equation can be factored into an equivalent equation<ref>Template:Cite book Extract of page 360</ref> <math display=block>ax^2+bx+c=a(x-r)(x-s)=0</math> where Template:Mvar and Template:Mvar are the solutions for Template:Mvar.

The quadratic formula <math display=block>x=\frac{-b\pm\sqrt{b^2-4ac} }{2a}</math> expresses the solutions in terms of Template:Mvar, Template:Mvar, and Template:Mvar. Completing the square is one of several ways for deriving the formula.

Solutions to problems that can be expressed in terms of quadratic equations were known as early as 2000 BC.<ref>Template:Cite book Extract of page 37</ref><ref>Template:Cite book Extract of page 26</ref>

Because the quadratic equation involves only one unknown, it is called "univariate". The quadratic equation contains only powers of Template:Math that are non-negative integers, and therefore it is a polynomial equation. In particular, it is a second-degree polynomial equation, since the greatest power is two.

Solving the quadratic equation

File:Quadratic equation coefficients.png

Figure 1. Plots of quadratic function Template:Nowrap, varying each coefficient separately while the other coefficients are fixed (at values a = 1, b = 0, c = 0)

A quadratic equation whose coefficients are real numbers can have either zero, one, or two distinct real-valued solutions, also called roots. When there is only one distinct root, it can be interpreted as two roots with the same value, called a double root. When there are no real roots, the coefficients can be considered as complex numbers with zero imaginary part, and the quadratic equation still has two complex-valued roots, complex conjugates of each-other with a non-zero imaginary part. A quadratic equation whose coefficients are arbitrary complex numbers always has two complex-valued roots which may or may not be distinct.

The solutions of a quadratic equation can be found by several alternative methods.

Factoring by inspection

It may be possible to express a quadratic equation Template:Math as a product Template:Math. 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 Template:Math or Template:Math. 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.<ref name=Washington2000>Template:Cite book</ref>Template:Rp If one is given a quadratic equation in the form Template:Math, the sought factorization has the form Template:Math, and one has to find two numbers Template:Math and Template:Math that add up to Template:Math and whose product is Template:Math (this is sometimes called "Vieta's rule"<ref>Template:Citation.</ref> and is related to Vieta's formulas). As an example, Template:Math factors as Template:Math. The more general case where Template:Math does not equal Template:Math 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 Template:Math or Template:Math, 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.<ref name=Washington2000/>Template:Rp

Completing the square

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

File:Polynomialdeg2.svg

Figure 2. For the quadratic function Template:Math, the points where the graph crosses the Template:Math-axis, Template:Math and Template:Math, are the solutions of the quadratic equation Template:Math.

The process of completing the square makes use of the algebraic identity <math display="block">x^2+2hx+h^2 = (x+h)^2,</math> which represents a well-defined algorithm that can be used to solve any quadratic equation.<ref name=Washington2000/>Template:Rp Starting with a quadratic equation in standard form, Template:Math

1. Divide each side by Template:Math, the coefficient of the squared term.
2. Subtract the constant term Template:Math from both sides.
3. Add the square of one-half of Template:Math, the coefficient of Template:Math, to both sides. This "completes the square", converting the left side into a perfect square.
4. Write the left side as a square and simplify the right side if necessary.
5. Produce two linear equations by equating the square root of the left side with the positive and negative square roots of the right side.
6. Solve each of the two linear equations.

We illustrate use of this algorithm by solving Template:Math <math display="block">2x^2+4x-4=0</math> <math display="block"> \ x^2+2x-2=0

</math>

<math display="block"> \ x^2+2x=2</math> <math display="block"> \ x^2+2x+1=2+1</math> <math display="block"> \left(x+1 \right)^2=3</math> <math display="block"> \ x+1=\pm\sqrt{3}</math> <math display="block"> \ x=-1\pm\sqrt{3}</math>

The plus–minus symbol "±" indicates that both <math display=inline>x=-1+\sqrt{3}</math> and <math display=inline>x=-1-\sqrt{3}</math> are solutions of the quadratic equation.<ref>Template:Citation</ref>

Quadratic formula and its derivation

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} Completing the square can be used to derive a general formula for solving quadratic equations, called the quadratic formula.<ref>Template:Citation, Chapter 13 §4.4, p. 291</ref> The mathematical proof will now be briefly summarized.<ref>Himonas, Alex. Calculus for Business and Social Sciences, p. 64 (Richard Dennis Publications, 2001).</ref> It can easily be seen, by polynomial expansion, that the following equation is equivalent to the quadratic equation: <math display="block">\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}.</math> Taking the square root of both sides, and isolating Template:Math, gives: <math display="block">x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.</math>

Some sources, particularly older ones, use alternative parameterizations of the quadratic equation such as Template:Math or Template:Math ,<ref name="kahan">Template:Citation</ref> where Template:Math 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, provides the same roots via the equation <math display="block">x = \frac{2c}{-b \pm \sqrt {b^2-4ac}}.</math> This can be deduced from the standard quadratic formula by Vieta's formulas, which assert that the product of the roots is Template:Math. It also follows from dividing the quadratic equation by <math>x^2</math> giving <math>cx^{-2}+bx^{-1}+a=0,</math> solving this for <math>x^{-1},</math> and then inverting.

One property of this form is that it yields one valid root when Template:Math, while the other root contains division by zero, because when Template:Math, 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 Template:Math for the other root. On the other hand, when Template:Math, the more common formula yields two correct roots whereas this form yields the zero root and an indeterminate form Template:Math.

When neither Template:Mvar nor Template:Mvar is zero, the equality between the standard quadratic formula and Muller's method, <math display="block">\frac{2c}{-b - \sqrt {b^2-4ac}} = \frac{-b + \sqrt {b^2-4ac}}{2a}\,,</math> can be verified by cross multiplication, and similarly for the other choice of signs.

Reduced quadratic equation

It is sometimes convenient to reduce a quadratic equation so that its leading coefficient is one. This is done by dividing both sides by Template:Math, which is always possible since Template:Math is non-zero. This produces the reduced quadratic equation:<ref>Alenit͡syn, Aleksandr and Butikov, Evgeniĭ. Concise Handbook of Mathematics and Physics, p. 38 (CRC Press 1997)</ref>

<math display="block">x^2+px+q=0,</math>

where Template:Math and Template:Math. This monic polynomial 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 <math display="block">x = - \frac{p}{2} \pm \sqrt{\left(\frac{p}{2}\right)^2 - q}\,.</math>

Discriminant

File:Quadratic eq discriminant.svg

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 Template:Math or an upper case Greek delta:<ref>Δ is the initial of the Greek word Διακρίνουσα, Diakrínousa, discriminant.</ref> <math display="block">\Delta = b^2 - 4ac.</math> 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 <math display="block">\frac{-b + \sqrt {\Delta}}{2a} \quad\text{and}\quad \frac{-b - \sqrt {\Delta}}{2a},</math> 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 <math>-\frac{b}{2a},</math> sometimes called a repeated or double root or two equal roots.
• If the discriminant is negative, then there are no real roots. Rather, there are two distinct (non-real) complex roots<ref>Template:Cite book</ref><math display="block">

-\frac{b}{2a} + i \frac{\sqrt {-\Delta}}{2a} \quad\text{and}\quad -\frac{b}{2a} - i \frac{\sqrt {-\Delta}}{2a}, </math> which are complex conjugates of each other. In these expressions Template:Math 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

Template:Quadratic function graph complex roots.svg The function Template:Math is a quadratic function.<ref>Template:Cite book</ref> 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 Template:Math, Template:Math, and Template:Math. If Template:Math, the parabola has a minimum point and opens upward. If Template:Math, the parabola has a maximum point and opens downward. The extreme point of the parabola, whether minimum or maximum, corresponds to its vertex. The Template:Math-coordinate of the vertex will be located at <math>\scriptstyle x=\tfrac{-b}{2a}</math>, and the Template:Math-coordinate of the vertex may be found by substituting this Template:Math-value into the function. The Template:Math-intercept is located at the point Template:Math.

The solutions of the quadratic equation Template:Math correspond to the roots of the function Template:Math, since they are the values of Template:Math for which Template:Math. If Template:Math, Template:Math, and Template:Math are real numbers and the domain of Template:Math is the set of real numbers, then the roots of Template:Math are exactly the Template:Math-coordinates of the points where the graph touches the Template:Math-axis. If the discriminant is positive, the graph touches the [[x-axis|Template:Math-axis]] at two points; if zero, the graph touches at one point; and if negative, the graph does not touch the Template:Math-axis.

Quadratic factorization

The term <math display="block">x - r</math> is a factor of the polynomial <math display="block">ax^2+bx+c</math> if and only if Template:Math is a root of the quadratic equation <math display="block">ax^2+bx+c=0.</math> It follows from the quadratic formula that <math display="block">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).</math> In the special case Template:Math where the quadratic has only one distinct root (i.e. the discriminant is zero), the quadratic polynomial can be factored as <math display="block">ax^2+bx+c = a \left( x + \frac{b}{2a} \right)^2.</math>

Graphical solution

File:Graphical calculation of root of quadratic equation.png

Figure 4. Graphing calculator computation of one of the two roots of the quadratic equation Template:Math. Although the display shows only five significant figures of accuracy, the retrieved value of Template:Math is 0.732050807569, accurate to twelve significant figures.

File:Visual.complex.root.finding.png

A quadratic function without real root: Template:Nowrap. The "3" is the imaginary part of the x-intercept. The real part is the x-coordinate of the vertex. Thus the roots are Template:Nowrap.

The solutions of the quadratic equation <math display="block">ax^2+bx+c=0</math> may be deduced from the graph of the quadratic function <math display="block">f(x)=ax^2+bx+c,</math> which is a parabola.

If the parabola intersects the Template:Mvar-axis in two points, there are two real roots, which are the Template:Mvar-coordinates of these two points (also called Template:Mvar-intercept).

If the parabola is tangent to the Template:Mvar-axis, there is a double root, which is the Template:Mvar-coordinate of the contact point between the graph and parabola.

If the parabola does not intersect the Template:Mvar-axis, there are two complex conjugate roots. Although these roots cannot be visualized on the graph, their real and imaginary parts can be.<ref name = "Norton1984">Template:Citation</ref>

Let Template:Mvar and Template:Mvar be respectively the Template:Mvar-coordinate and the Template:Mvar-coordinate of the vertex of the parabola (that is the point with maximal or minimal Template:Mvar-coordinate. The quadratic function may be rewritten <math display="block"> y = a(x - h)^2 + k.</math> Let Template:Mvar be the distance between the point of Template:Mvar-coordinate Template:Math on the axis of the parabola, and a point on the parabola with the same Template:Mvar-coordinate (see the figure; there are two such points, which give the same distance, because of the symmetry of the parabola). Then the real part of the roots is Template:Mvar, and their imaginary part are Template:Math. That is, the roots are <math display="block">h+id \quad \text{and} \quad h-id,</math> or in the case of the example of the figure <math display="block">5+3i \quad \text{and} \quad 5-3i.</math>

Avoiding loss of significance

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 Template:Math and Template:Math 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, Template:Math, can be computed as <math>(c/a)/R</math> where Template:Math is the root that is bigger in magnitude. This is equivalent to using the formula

<math display="block">x =\frac{-2c}{b \pm \sqrt {b^2-4ac}}</math>

using the plus sign if <math>b>0</math> and the minus sign if <math>b<0.</math>

A second form of cancellation can occur between the terms Template:Math and Template:Math 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.<ref name="kahan"/><ref name="Higham2002">Template:Citation</ref>

Examples and applications

File:La Jolla Cove cliff diving - 02.jpg

The trajectory of the cliff jumper is parabolic because horizontal displacement is a linear function of time <math>x=v_x t</math>, while vertical displacement is a quadratic function of time <math>y=\tfrac{1}{2} at^2+v_y t+h</math>. As a result, the path follows quadratic equation <math>y=\tfrac{a}{2v_x^2} x^2+\tfrac{v_y}{v_x} x+h</math>, where <math>v_x</math> and <math>v_y</math> are horizontal and vertical components of the original velocity, Template:Math is gravitational acceleration and Template:Math is original height. The Template:Math value should be considered negative here, as its direction (downwards) is opposite to the height measurement (upwards).

The golden ratio is found as the positive solution of the quadratic equation <math>x^2-x-1=0.</math>

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.

Critical points of a cubic function and inflection points of a quartic function are found by solving a quadratic equation.

In physics, for motion with constant acceleration <math>a</math>, the displacement or position <math>x</math> of a moving body can be expressed as a quadratic function of time <math>t</math> given the initial position <math>x_0</math> and initial velocity <math>v_0</math>: <math display="inline">x = x_0 + v_0 t + \frac{1}2 at^2</math>.

In chemistry, the pH of a solution of weak acid can be calculated from the negative base-10 logarithm of the positive root of a quadratic equation in terms of the acidity constant and the analytical concentration of the acid.

History

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.<ref name=Friberg2009>Template:Cite journal</ref> In modern notation, the problems typically involved solving a pair of simultaneous equations of the form: <math display="block"> x+y=p,\ \ xy=q, </math> which is equivalent to the statement that Template:Mvar and Template:Mvar are the roots of the equation:<ref name=Stillwell2004>Template:Cite book</ref>Template:Rp <math display="block">z^2+q=pz.</math>

The steps given by Babylonian scribes for solving the above rectangle problem, in terms of Template:Mvar and Template:Mvar, were as follows:

1. Compute half of p.
2. Square the result.
3. Subtract q.
4. Find the (positive) square root using a table of squares.
5. Add together the results of steps (1) and (4) to give Template:Math.

In modern notation this means calculating <math>x = \frac{p}{2} + \sqrt{\left(\frac{p}{2}\right)^2 - q}</math>, which is equivalent to the modern day quadratic formula for the larger real root (if any) <math>x = \frac{-b + \sqrt{b^2 - 4ac}}{2a}</math> with Template:Math, Template:Math, and Template:Math.

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.<ref>Template:Cite book</ref> 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.<ref name=Henderson>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref name=Aitken>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Rules for quadratic equations were given in The Nine Chapters on the Mathematical Art, a Chinese treatise on mathematics.<ref name=Aitken/><ref>Template:Cite book</ref> 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.<ref>Template:Cite book Extract of page 134</ref>

In 628 AD, Brahmagupta, an Indian mathematician, gave in his book Brāhmasphuṭasiddhānta the first explicit (although still not completely general) solution of the quadratic equation Template:Math 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."<ref>Brāhmasphuṭasiddhānta, Colebrook translation, 1817, page 346; cited by Template:Cite book</ref> This is equivalent to <math display="block">x = \frac{\sqrt{4ac+b^2}-b}{2a}.</math> The Bakhshali Manuscript written in India in the 7th century AD contained an algebraic formula for solving quadratic equations, as well as linear indeterminate equations (originally of type Template:Math). Muhammad ibn Musa al-Khwarizmi (9th century) 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.<ref name=Katz2007>Template:Cite journal</ref> He also described the method of completing the square and recognized that the discriminant must be positive,<ref name=Katz2007/><ref name=Boyer1991/>Template:Rp 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.<ref name=Boyer1991>Template:Cite book</ref>Template:Rp While al-Khwarizmi himself did not accept negative solutions, later Islamic mathematicians that succeeded him accepted negative solutions,<ref name=Katz2007/>Template:Rp as well as irrational numbers as solutions.<ref>Template:MacTutor "Algebra was a unifying theory which allowed rational numbers, irrational numbers, geometrical magnitudes, etc., to all be treated as "algebraic objects"."</ref> 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.<ref>Jacques Sesiano, "Islamic mathematics", p. 148, in Template:Citation</ref> The 9th century Indian mathematician Sridhara wrote down rules for solving quadratic equations.<ref>Template:Cite book</ref>

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.<ref name=Livio2006>Template:Cite book</ref> His solution was largely based on Al-Khwarizmi's work.<ref name=Katz2007/> 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.<ref name=Ron>Template:Cite book</ref> 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.<ref>Template:Citation</ref> In 1637 René Descartes published La Géométrie containing the quadratic formula in the form we know today.

Advanced topics

Alternative methods of root calculation

Vieta's formulas

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

Vieta's formulas (named after François Viète) are the relations <math display="block"> x_1 + x_2 = -\frac{b}{a}, \quad x_1 x_2 = \frac{c}{a}</math> between the roots of a quadratic polynomial and its coefficients. They result from comparing term by term the relation <math display="block">\left( x - x_1 \right) \left( x-x_2 \right ) = x^2 - \left( x_1+x_2 \right)x +x_1 x_2 = 0</math> with the equation <math display="block"> x^2 + \frac ba x +\frac ca = 0.</math>

The first Vieta's formula is useful for graphing a quadratic function. Since the graph is symmetric with respect to a vertical line through the vertex, the vertex's Template:Math-coordinate is located at the average of the roots (or intercepts). Thus the Template:Math-coordinate of the vertex is <math display="block"> x_V = \frac {x_1 + x_2} {2} = -\frac{b}{2a}.</math> The Template:Math-coordinate can be obtained by substituting the above result into the given quadratic equation, giving <math display="block"> y_V = - \frac{b^2}{4a} + c = - \frac{ b^2 - 4ac} {4a}.</math> Also, these formulas for the vertex can be deduced directly from the formula (see Completing the square) <math display="block">ax^2+bx+c=a \left(x+\frac b{2a}\right)^2 - \frac{b^2-4ac}{4a}.</math>

For numerical computation, Vieta's formulas provide a useful method for finding the roots of a quadratic equation in the case where one root is much smaller than the other. If Template:Math, then Template:Math, and we have the estimate: <math display="block"> x_1 \approx -\frac{b}{a} .</math> The second Vieta's formula then provides: <math display="block">x_2 = \frac{c}{a x_1} \approx -\frac{c}{b} .</math> 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 Template:Math), which causes round-off error in a numerical evaluation. The figure shows the difference betweenTemplate:Clarify (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 Template:Math increases, initially the quadratic formula is accurate, and the approximate formula improves in accuracy, leading to a smaller difference between the methods as Template:Math 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

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.<ref name=Ballew2007>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> 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,

where the sign of the ± symbol is chosen so that Template:Math and Template:Math may both be positive. By substituting

Template:NumBlk \tan\theta </math>|Template:EquationRef}}

and then multiplying through by Template:Math, we obtain

Template:NumBlk \sin\theta \cos\theta \pm \cos^2\theta = 0 .</math>|Template:EquationRef}}

Introducing functions of Template:Math and rearranging, we obtain

Template:NumBlk{b} ,</math>|Template:EquationRef}}

Template:NumBlk{b} ,</math>|Template:EquationRef}}

where the subscripts Template:Math and Template:Math correspond, respectively, to the use of a negative or positive sign in equation Template:EquationNote. Substituting the two values of Template:Math or Template:Math found from equations Template:EquationNote or Template:EquationNote into Template:EquationNote gives the required roots of Template:EquationNote. Complex roots occur in the solution based on equation Template:EquationNote if the absolute value of Template:Math 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.<ref name=Seares1945>Template:Cite journal</ref> Calculating complex roots would require using a different trigonometric form.<ref name=Aude1938>Template:Cite journal</ref>

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: <math display="block">4.16130x^2 + 9.15933x - 11.4207 = 0</math>

1. 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.
2. <math>\log a = 0.6192290, \log b = 0.9618637, \log c = 1.0576927</math>
3. <math>2 \sqrt{ac}/b = 2 \times 10^{(0.6192290 + 1.0576927)/2 - 0.9618637} = 1.505314 </math>
4. <math>\theta = (\tan^{-1}1.505314) / 2 = 28.20169^{\circ} \text{ or } -61.79831^{\circ} </math>
5. <math>\log | \tan \theta | = -0.2706462 \text{ or } 0.2706462</math>
6. <math> \log{\textstyle \sqrt{c/a}} = (1.0576927 - 0.6192290) / 2 = 0.2192318</math>
7. <math>x_1 = 10^{0.2192318 - 0.2706462} = 0.888353</math> (rounded to six significant figures) <math display="block">x_2 = -10^{0.2192318 + 0.2706462} = -3.08943</math>

Solution for complex roots in polar coordinates

If the quadratic equation <math>ax^2+bx+c=0</math> with real coefficients has two complex roots—the case where <math>b^2-4ac<0,</math> 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<ref>Simons, Stuart, "Alternative approach to complex roots of real quadratic equations", Mathematical Gazette 93, March 2009, 91–92.</ref>

<math display="block">x_1, \, x_2=r(\cos \theta \pm i\sin \theta), </math>

where <math>r=\sqrt{\tfrac{c}{a}}</math> and <math>\theta =\cos ^{-1}\left(\tfrac{-b}{2\sqrt{ac}}\right).</math>

Geometric solution

File:LillsQuadratic.svg

Figure 6. Geometric solution of Template:Math 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 Template:Math, Template:Math, Template:Math 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 Template:Math or SA. If Template:Math is Template:Math the coefficients may be read off directly. Thus the solutions in the diagram are −AX1/SA and −AX2/SA.<ref>Template:Citation</ref>

File:CarlyleCircle.svg

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.<ref name=Wolfram>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> Carlyle circles have been used to develop ruler-and-compass constructions of regular polygons.

Generalization of quadratic equation

The formula and its derivation remain correct if the coefficients Template:Math, Template:Math and Template:Math are complex numbers, or more generally members of any field whose characteristic is not Template:Math. (In a field of characteristic 2, the element Template:Math is zero and it is impossible to divide by it.)

The symbol <math display="block">\pm \sqrt {b^2-4ac}</math> in the formula should be understood as "either of the two elements whose square is Template:Math, 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 Template:Math. 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

In a field of characteristic Template:Math, the quadratic formula, which relies on Template:Math being a unit, does not hold. Consider the monic quadratic polynomial <math display="block">x^{2} + bx + c</math> over a field of characteristic Template:Math. If Template:Math, then the solution reduces to extracting a square root, so the solution is <math display="block">x = \sqrt{c}</math> and there is only one root since <math display="block">-\sqrt{c} = -\sqrt{c} + 2\sqrt{c} = \sqrt{c}.</math> In summary, <math display="block">\displaystyle x^{2} + c = (x + \sqrt{c})^{2}.</math> See quadratic residue for more information about extracting square roots in finite fields.

In the case that Template:Math, 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 Template:Math of Template:Math to be a root of the polynomial Template:Math, an element of the splitting field of that polynomial. One verifies that Template:Math is also a root. In terms of the 2-root operation, the two roots of the (non-monic) quadratic Template:Math are <math display="block">\frac{b}{a}R\left(\frac{ac}{b^2}\right)</math> and <math display="block">\frac{b}{a}\left(R\left(\frac{ac}{b^2}\right)+1\right).</math>

For example, let Template:Math denote a multiplicative generator of the group of units of Template:Math, the Galois field of order four (thus Template:Math and Template:Math are roots of Template:Math over Template:Math. Because Template:Math, Template:Math is the unique solution of the quadratic equation Template:Math. On the other hand, the polynomial Template:Math is irreducible over Template:Math, but it splits over Template:Math, where it has the two roots Template:Math and Template:Math, where Template:Math is a root of Template:Math in Template:Math.

This is a special case of Artin–Schreier theory.

See also

• Solving quadratic equations with continued fractions
• Linear equation
• Cubic function
• Quartic equation
• Quintic equation
• Fundamental theorem of algebra

External links

• Template:Springer
• {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web

|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:QuadraticEquation%7CQuadraticEquation.html}} |title = Quadratic equations |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}

• 101 uses of a quadratic equation Template:Webarchive
• 101 uses of a quadratic equation: Part II Template:Webarchive