In mathematics, a quadratic equation () is an equation that can be rearranged in standard form as Extract of page 219
where the variable represents an unknown number, and , , and represent known numbers, where . (If and then the equation is linear, not quadratic.) The numbers , , and are the coefficients of the equation and may be distinguished by calling them the quadratic coefficient, the linear coefficient and the constant coefficient or free term, respectively.Protters & Morrey: "Calculus and Analytic Geometry. First Course".
The values of 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 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 Extract of page 360
where and are the solutions for .
The quadratic formula
expresses the solutions in terms of , , and . 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. Extract of page 37 Extract of page 26
The quadratic equation contains only powers of 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
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 as a product . 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 or . 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. If one is given a quadratic equation in the form , the sought factorization has the form , and one has to find two numbers and that add up to and whose product is (this is sometimes called "Vieta's rule". and is related to Vieta's formulas). As an example, factors as . The more general case where does not equal 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 or , 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.
Completing the square
The process of completing the square makes use of the algebraic identity
which represents a well-defined algorithm that can be used to solve any quadratic equation. Starting with a quadratic equation in standard form, Divide each side by , the coefficient of the squared term. Subtract the constant term from both sides. Add the square of one-half of , the coefficient of , 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 each of the two linear equations.
We illustrate use of this algorithm by solving
The plus–minus symbol "±" indicates that both and are solutions of the quadratic equation.
Quadratic formula and its derivation
Completing the square can be used to derive a general formula for solving quadratic equations, called the quadratic formula., Chapter 13 §4.4, p. 291 The mathematical proof will now be briefly summarized.Himonas, Alex. Calculus for Business and Social Sciences, p. 64 (Richard Dennis Publications, 2001). It can easily be seen, by polynomial expansion, that the following equation is equivalent to the quadratic equation:
Taking the square root of both sides, and isolating , gives:
Some sources, particularly older ones, use alternative parameterizations of the quadratic equation such as or , where 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
This can be deduced from the standard quadratic formula by Vieta's formulas, which assert that the product of the roots is . It also follows from dividing the quadratic equation by giving solving this for and then inverting.
One property of this form is that it yields one valid root when , while the other root contains division by zero, because when , 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 for the other root. On the other hand, when , the more common formula yields two correct roots whereas this form yields the zero root and an indeterminate form .
When neither nor is zero, the equality between the standard quadratic formula and Muller's method,
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 , which is always possible since is non-zero. This produces the reduced quadratic equation:Alenit͡syn, Aleksandr and Butikov, Evgeniĭ. Concise Handbook of Mathematics and Physics, p. 38 (CRC Press 1997)
where and . 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
Discriminant
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 or an upper case Greek delta:Δ is the initial of the Greek word Διακρίνουσα, Diakrínousa, discriminant.
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 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 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 which are complex conjugates of each other. In these expressions 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
The function is a quadratic function. 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 , , and . If , the parabola has a minimum point and opens upward. If , the parabola has a maximum point and opens downward. The extreme point of the parabola, whether minimum or maximum, corresponds to its vertex. The -coordinate of the vertex will be located at , and the -coordinate of the vertex may be found by substituting this -value into the function. The -intercept is located at the point .
The solutions of the quadratic equation correspond to the roots of the function , since they are the values of for which . If , , and are real numbers and the domain of is the set of real numbers, then the roots of are exactly the -coordinates of the points where the graph touches the -axis. If the discriminant is positive, the graph touches the -axis at two points; if zero, the graph touches at one point; and if negative, the graph does not touch the -axis.
Quadratic factorization The term
is a factor of the polynomial
if and only if is a root of the quadratic equation
It follows from the quadratic formula that
In the special case where the quadratic has only one distinct root (i.e. the discriminant is zero), the quadratic polynomial can be factored as
Graphical solution
The solutions of the quadratic equation
may be obtained from the graph of the quadratic function
which is a parabola.
If the parabola intersects the -axis in two points, there are two real roots, which are the -coordinates of these two points (also called -intercept).
If the parabola is tangent to the -axis, there is a double root, which is the -coordinate of the contact point between the graph and parabola.
If the parabola does not intersect the -axis, there are two complex conjugate roots. Although these roots cannot be visualized on the graph, their real and imaginary parts can be.
Let and be respectively the -coordinate and the -coordinate of the vertex of the parabola (that is the point with maximal or minimal -coordinate. The quadratic function may be rewritten
Let be the distance between the point of -coordinate on the axis of the parabola, and a point on the parabola with the same -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 , and their imaginary part are . That is, the roots are
or in the case of the example of the figure
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 and 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, , can be computed as where is the root that is bigger in magnitude. This is equivalent to using the formula
using the plus sign if and the minus sign if