Quadratic formula

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A quadratic function with roots x = 1 and x = 4. Quadratic roots.svg
A quadratic function with roots x = 1 and x = 4.

In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, grouping, AC method), completing the square, graphing and others. [1]

Contents

Given a general quadratic equation of the form

with x representing an unknown, a, b and c representing constants with a ≠ 0, the quadratic formula is:

where the plus-minus symbol "±" indicates that the quadratic equation has two solutions. [2] Written separately, they become:

Each of these two solutions is also called a root (or zero) of the quadratic equation. Geometrically, these roots represent the x-values at which any parabola, explicitly given as y = ax2 + bx + c, crosses the x-axis. [3]

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, [4] and the number of real zeros the quadratic equation contains. [5]

Equivalent formulations

The quadratic formula may also be written as

which may be simplified to

The expression inside the square root is called the discriminant.
This version of the formula makes it easy to find the roots when using a calculator.
The version above is also convenient when complex roots are involved, in which case the expression outside the square root is the real part and the square root expression is the imaginary part:

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 is

Some sources, particularly older ones, use alternative parameterizations of the quadratic equation such as

, where , [6]

or

, where . [7]

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. [8] [9] [10] [11] 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 c/a 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:

  1. Divide each side by to make the polynomial monic.
  2. Rearrange.
  3. Add to both sides to complete the square.
  4. Rearrange the terms in the right side to have a common denominator.
  5. Take the square root of both sides.
  6. Isolate .

Completing the square can also be accomplished by a sometimes shorter and simpler sequence: [12]

  1. Multiply each side by ,
  2. Rearrange.
  3. Add to both sides to complete the square.
  4. Take the square root of both sides.
  5. Isolate .

In which case, the quadratic formula can also be derived as follows:

This derivation of the quadratic formula is ancient and was known in India at least as far back as 1025. [13] 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. [12]

Method 3

Similar to Method 1, divide each side by to make left-hand-side polynomial monic (i.e., the coefficient of becomes 1).

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:

or

By re-expressing and back into and , respectively, the quadratic formula can then be obtained.[ citation needed ]

By substitution

Another technique is solution by substitution. [14] 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 (to move it to the right hand side) 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: [15]

Let the roots of the standard quadratic equation be r1 and r2. The derivation starts by recalling the identity:

Taking the square root on both sides, we get:

Since the coefficient a ≠ 0, we can divide the standard equation by a 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 b/a, and the product of those roots is given by c/a. Hence the identity can be rewritten as:

Now,

Since r2 = −r1b/a, 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, [16] which is an early part of Galois theory. [17] 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 p = −(α + β) and q = αβ.

Since the order of multiplication does not matter, one can switch α and β and the values of p and q will not change: one can say that p and q are symmetric polynomials in α and β. In fact, they are the elementary symmetric polynomials – any symmetric polynomial in α and β can be expressed in terms of α + β and αβ 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 n is related to the ways of rearranging ("permuting") n terms, which is called the symmetric group on n letters, and denoted Sn. For the quadratic polynomial, the only way to rearrange two terms is to swap them ("transpose" them), and thus solving a quadratic polynomial is simple.

To find the roots α and β, 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 r1 = α + β is a symmetric function in α and β, so it can be expressed in terms of p and q, and in fact r1 = −p as noted above. But r2 = αβ is not symmetric, since switching α and β yields r2 = βα (formally, this is termed a group action of the symmetric group of the roots). Since r2 is not symmetric, it cannot be expressed in terms of the coefficients p and q, as these are symmetric in the roots and thus so is any polynomial expression involving them. Changing the order of the roots only changes r2 by a factor of −1, and thus the square r22 = (αβ)2 is symmetric in the roots, and thus expressible in terms of p and q. 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 p = b/a, q = c/a yields the usual form for when a quadratic is not monic. The resolvents can be recognized as r1/2 = p/2 = b/2a being the vertex, and r22 = p2 − 4q is the discriminant (of a monic polynomial).

A similar but more complicated method works for cubic equations, where one has three resolvents and a quadratic equation (the "resolving polynomial") relating r2 and r3, which one can solve by the quadratic equation, and similarly for a quartic equation (degree 4), whose resolving polynomial is a cubic, which can in turn be solved. [16] 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. [18] The Egyptian Berlin Papyrus, dating back to the Middle Kingdom (2050 BC to 1650 BC), contains the solution to a two-term quadratic equation. [19]

The Greek mathematician Euclid (circa 300 BC) used geometric methods to solve quadratic equations in Book 2 of his Elements , an influential mathematical treatise. [20] Rules for quadratic equations appear in the Chinese The Nine Chapters on the Mathematical Art circa 200 BC. [21] [22] In his work Arithmetica , the Greek mathematician Diophantus (circa 250 AD) solved quadratic equations with a method more recognizably algebraic than the geometric algebra of Euclid. [20] His solution gives only one root, even when both roots are positive. [23]

The Indian mathematician Brahmagupta (597–668 AD) explicitly described the quadratic formula in his treatise Brāhmasphuṭasiddhānta published in 628 AD, [24] but written in words instead of symbols. [25] His solution of the quadratic equation ax2 + bx = c was 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." [26] This is equivalent to:

The 9th-century Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī solved quadratic equations algebraically. [27] The quadratic formula covering all cases was first obtained by Simon Stevin in 1594. [28] In 1637 René Descartes published La Géométrie containing special cases of the quadratic formula in the form we know today. [29]

Significant uses

Geometric significance

Graph of y = ax + bx + c, where a and the discriminant b - 4ac are positive, with
Roots and y-intercept in red
Vertex and axis of symmetry in blue
Focus and directrix in pink Quadratic function graph key values.svg
Graph of y = ax + bx + c, where a and the discriminant b − 4ac are positive, with
  • Roots and y-intercept in red
  • Vertex and axis of symmetry in blue
  • Focus and directrix in pink

In terms of coordinate geometry, a parabola is a curve whose (x, y)-coordinates are described by a second-degree polynomial, i.e. any equation of the form:

where p represents the polynomial of degree 2 and a0, a1, and a2 ≠ 0 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 x-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 x = −b/2a. The other term, b2 − 4ac/2a, 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 x value of the only zero, that is, there is only one possible solution to the quadratic equation. Algebraically, this means that b2 − 4ac = 0, or simply b2 − 4ac = 0 (where the left-hand side is referred to as the discriminant). 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 i, where i = −1  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 x where the parabola crosses the x-axis.

Dimensional analysis

If the constants a, b, and/or c are not unitless, then the units of x must be equal to the units of b/a, due to the requirement that ax2 and bx agree on their units. Furthermore, by the same logic, the units of c must be equal to the units of b2/a, which can be verified without solving for x. This can be a powerful tool for verifying that a quadratic expression of physical quantities has been set up correctly, prior to solving this.

See also

Related Research Articles

Complex number Element of a number system in which –1 has a square root

In mathematics, a complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a symbol called the imaginary unit, and satisfying the equation i2 = −1. Because no "real" number satisfies this equation, i was called an imaginary number by René Descartes. For the complex number a + bi, a is called the real part and b is called the imaginary part. The set of complex numbers is denoted by either of the symbols or C. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world.

In algebra, a quadratic equation is any equation that can be rearranged in standard form as

In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and determines various properties of the roots. It is generally defined as a polynomial function of the coefficients of the original polynomial. The discriminant is widely used in polynomial factoring, number theory, and algebraic geometry. It is often denoted by the symbol .

Factorization (Mathematical) decomposition into a product

In mathematics, factorization or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. For example, 3 × 5 is a factorization of the integer 15, and (x – 2)(x + 2) is a factorization of the polynomial x2 – 4.

Cubic equation Polynomial equation of degree 3

In algebra, a cubic equation in one variable is an equation of the form

Quadratic function Polynomial function of degree two

In algebra, a quadratic function, a quadratic polynomial, a polynomial of degree 2, or simply a quadratic, is a polynomial function with one or more variables in which the highest-degree term is of the second degree.

Quartic equation Polynomial equation

In mathematics, a quartic equation is one which can be expressed as a quartic function equaling zero. The general form of a quartic equation is

Quintic function Polynomial function of degree 5

In algebra, a quintic function is a function of the form

Completing the square Method for solving quadratic equations

In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form

In mathematics, a quadratic irrational number is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible over the rational numbers. Since fractions in the coefficients of a quadratic equation can be cleared by multiplying both sides by their common denominator, a quadratic irrational is an irrational root of some quadratic equation whose coefficients are integers. The quadratic irrational numbers, a subset of the complex numbers, are algebraic numbers of degree 2, and can therefore be expressed as

Quartic function Polynomial function of degree four

In algebra, a quartic function is a function of the form

In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials: the Hermite polynomials, Laguerre polynomials, Jacobi polynomials.

In mathematics, an algebraic equation or polynomial equation is an equation of the form

In mathematics, the field trace is a particular function defined with respect to a finite field extension L/K, which is a K-linear map from L onto K.

This article describes periodic points of some complex quadratic maps. A map is a formula for computing a value of a variable based on its own previous value or values; a quadratic map is one that involves the previous value raised to the powers one and two; and a complex map is one in which the variable and the parameters are complex numbers. A periodic point of a map is a value of the variable that occurs repeatedly after intervals of a fixed length.

In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional power. Examples of such functions are:

Eisenstein integer

In mathematics, Eisenstein integers, occasionally also known as Eulerian integers, are complex numbers of the form

In algebra, a nested radical is a radical expression that contains (nests) another radical expression. Examples include

In algebra, casus irreducibilis is one of the cases that may arise in attempting to solve polynomials of degree 3 or higher with integer coefficients, to obtain roots that are expressed with radicals. It shows that many algebraic numbers are real-valued but cannot be expressed in radicals without introducing complex numbers. The most notable occurrence of casus irreducibilis is in the case of cubic polynomials that are irreducible over the rational numbers and have three real roots, which was proven by Pierre Wantzel in 1843. One can decide whether a given irreducible cubic polynomial is in casus irreducibilis using the discriminant Δ, via Cardano's formula. Let the cubic equation be given by

Resolvent cubic

In algebra, a resolvent cubic is one of several distinct, although related, cubic polynomials defined from a monic polynomial of degree four:

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