Quadratic formula
In elementary algebra, the quadratic formula is a formula that provides the solution to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring, completing the square, graphing and others.
Given a general quadratic equation of the form
with representing an unknown,, and representing constants with, the quadratic formula is:
where the plus-minus symbol "±" indicates that the quadratic equation has two solutions. Written separately, they become:
Each of these two solutions is also called a root of the quadratic equation. Geometrically, these roots represent the -values at which any parabola, explicitly given as, crosses the -axis.
As well as being a formula that yields the zeros of any parabola, the quadratic formula can also be used to identify the axis of symmetry of the parabola, and the number of real zeros the quadratic equation contains.
Equivalent formulations
The quadratic formula may also be written as:which may be simplified to:
This version of the formula is convenient when complex roots are involved, in which case the expression outside the square root will be the real part—and the square root expression the imaginary part. The expression inside the square root is a discriminant.
Muller's method
A lesser known quadratic formula, which is used in Muller's method and which can be found from Vieta's formulas, provides the same roots via the equation:Formulations based on alternative parametrizations
The standard parametrization of the quadratic equation isSome sources, particularly older ones, use alternative parameterizations of the quadratic equation such as
or
These alternative parametrizations result in slightly different forms for the solution, but which are otherwise equivalent to the standard parametrization.
Derivations of the formula
Many different methods to derive the quadratic formula are available in the literature. The standard one is a simple application of the completing the square technique. Alternative methods are sometimes simpler than completing the square, and may offer interesting insight into other areas of mathematics.By using the 'completing the square' technique
Standard method
Divide the quadratic equation by, which is allowed because is non-zero:Subtract from both sides of the equation, yielding:
The quadratic equation is now in a form to which the method of completing the square is applicable. In fact, by adding a constant to both sides of the equation such that the left hand side becomes a complete square, the quadratic equation becomes:
which produces:
Accordingly, after rearranging the terms on the right hand side to have a common denominator, we obtain:
The square has thus been completed. Taking the square root of both sides yields the following equation:
In which case, isolating the would give the quadratic formula:
There are many alternatives of this derivation with minor differences, mostly concerning the manipulation of.
Method 2
The majority of algebra texts published over the last several decades teach completing the square using the sequence presented earlier:- Divide each side by to make the polynomial monic.
- Rearrange.
- Add to both sides to complete the square.
- Rearrange the terms in the right side to have a common denominator.
- Take the square root of both sides.
- Isolate '.
- Multiply each side by ',
- Rearrange.
- Add ' to both sides to complete the square.
- Take the square root of both sides.
- Isolate '.
This derivation of the quadratic formula is ancient and was known in India at least as far back as 1025. Compared with the derivation in standard usage, this alternate derivation avoids fractions and squared fractions until the last step and hence does not require a rearrangement after step 3 to obtain a common denominator in the right side.
Method 3
Similar to Method 1, divide each side by ' to make left-hand-side polynomial monic.Write the equation in a more compact and easier-to-handle format:
where ' and '.
Complete the square by adding ' to the first two terms and subtracting it from the third term:
Rearrange the left side into a difference of two squares:
and factor it:
which implies that either
or
Each of these two equations is linear and can be solved for ', obtaining:
orBy re-expressing ' and ' back into ' and , respectively, the quadratic formula can then be obtained.
By substitution
Another technique is solution by substitution. In this technique, we substitute ' into the quadratic to get:Expanding the result and then collecting the powers of ' produces:
We have not yet imposed a second condition on ' and ', so we now choose ' so that the middle term vanishes. That is, ' or '. Subtracting the constant term from both sides of the equation and then dividing by ' gives:
Substituting for ' gives:
Therefore,
By re-expressing ' in terms of ' using the formula ', the usual quadratic formula can then be obtained:
By using algebraic identities
The following method was used by many historical mathematicians:Let the roots of the standard quadratic equation be and. The derivation starts by recalling the identity:
Taking the square root on both sides, we get:
Since the coefficient, we can divide the standard equation by to obtain a quadratic polynomial having the same roots. Namely,
From this we can see that the sum of the roots of the standard quadratic equation is given by, and the product of those roots is given by.
Hence the identity can be rewritten as:
Now,
Since, if we take
then we obtain
and if we instead take
then we calculate that
Combining these results by using the standard shorthand ±, we have that the solutions of the quadratic equation are given by:
By Lagrange resolvents
An alternative way of deriving the quadratic formula is via the method of Lagrange resolvents, which is an early part of Galois theory.This method can be generalized to give the roots of cubic polynomials and quartic polynomials, and leads to Galois theory, which allows one to understand the solution of algebraic equations of any degree in terms of the symmetry group of their roots, the Galois group.
This approach focuses on the roots more than on rearranging the original equation. Given a monic quadratic polynomial
assume that it factors as
Expanding yields
where and.
Since the order of multiplication does not matter, one can switch and and the values of and will not change: one can say that and are symmetric polynomials in and. In fact, they are the elementary symmetric polynomials – any symmetric polynomial in and can be expressed in terms of and The Galois theory approach to analyzing and solving polynomials is: given the coefficients of a polynomial, which are symmetric functions in the roots, can one "break the symmetry" and recover the roots? Thus solving a polynomial of degree is related to the ways of rearranging terms, which is called the symmetric group on letters, and denoted. For the quadratic polynomial, the only way to rearrange two terms is to swap them, and thus solving a quadratic polynomial is simple.
To find the roots and, consider their sum and difference:
These are called the Lagrange resolvents of the polynomial; notice that one of these depends on the order of the roots, which is the key point. One can recover the roots from the resolvents by inverting the above equations:
Thus, solving for the resolvents gives the original roots.
Now is a symmetric function in and, so it can be expressed in terms of and, and in fact as noted above. But is not symmetric, since switching and yields . Since is not symmetric, it cannot be expressed in terms of the coefficients and, as these are symmetric in the roots and thus so is any polynomial expression involving them. Changing the order of the roots only changes by a factor of −1, and thus the square is symmetric in the roots, and thus expressible in terms of and. Using the equation
yields
and thus
If one takes the positive root, breaking symmetry, one obtains:
and thus
Thus the roots are
which is the quadratic formula. Substituting yields the usual form for when a quadratic is not monic. The resolvents can be recognized as being the vertex, and is the discriminant.
A similar but more complicated method works for cubic equations, where one has three resolvents and a quadratic equation relating and, which one can solve by the quadratic equation, and similarly for a quartic equation, whose resolving polynomial is a cubic, which can in turn be solved. The same method for a quintic equation yields a polynomial of degree 24, which does not simplify the problem, and, in fact, solutions to quintic equations in general cannot be expressed using only roots.
Historical development
The earliest methods for solving quadratic equations were geometric. Babylonian cuneiform tablets contain problems reducible to solving quadratic equations. The Egyptian Berlin Papyrus, dating back to the Middle Kingdom, contains the solution to a two-term quadratic equation.The Greek mathematician Euclid used geometric methods to solve quadratic equations in Book 2 of his Elements, an influential mathematical treatise. Rules for quadratic equations appear in the Chinese The Nine Chapters on the Mathematical Art circa 200 BC. In his work Arithmetica, the Greek mathematician Diophantus solved quadratic equations with a method more recognizably algebraic than the geometric algebra of Euclid. His solution gives only one root, even when both roots are positive.
The Indian mathematician Brahmagupta explicitly described the quadratic formula in his treatise Brāhmasphuṭasiddhānta published in 628 AD, but written in words instead of symbols. His solution of the quadratic equation was as follows: "To the absolute number multiplied by four times the square, add the square of the middle term; the square root of the same, less the middle term, being divided by twice the square is the value."
This is equivalent to:
The 9th-century Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī solved quadratic equations algebraically. The quadratic formula covering all cases was first obtained by Simon Stevin in 1594. In 1637 René Descartes published La Géométrie containing special cases of the quadratic formula in the form we know today.
Significant uses
Geometric significance
In terms of coordinate geometry, a parabola is a curve whose -coordinates are described by a second-degree polynomial, i.e. any equation of the form:where represents the polynomial of degree 2 and and are constant coefficients whose subscripts correspond to their respective term's degree. The geometrical interpretation of the quadratic formula is that it defines the points on the -axis where the parabola will cross the axis. Additionally, if the quadratic formula was looked at as two terms,
the axis of symmetry appears as the line. The other term,, gives the distance the zeros are away from the axis of symmetry, where the plus sign represents the distance to the right, and the minus sign represents the distance to the left.
If this distance term were to decrease to zero, the value of the axis of symmetry would be the value of the only zero, that is, there is only one possible solution to the quadratic equation. Algebraically, this means that, or simply . This is one of three cases, where the discriminant indicates how many zeros the parabola will have. If the discriminant is positive, the distance would be non-zero, and there will be two solutions. However, there is also the case where the discriminant is less than zero, and this indicates the distance will be imaginary or some multiple of the complex unit, where and the parabola's zeros will be complex numbers. The complex roots will be complex conjugates, where the real part of the complex roots will be the value of the axis of symmetry. There will be no real values of where the parabola crosses the -axis.