Solving Quadratic Equations

Quadratic equations are second degree polynomial equations, and they have the following standard form.

where

is the variable,

and

are constants, and

. Quadratic equations have been studied more than 4,000 years, dating back to Babylonian mathematicians who found the dimensions

and

of a rectangle with a given area

and perimeter

. See Figure 1.

In modern notation, the problem that the Babylonians formulated and solved is the following: Given

and

, find

and

such that

Combining

and

yields

, which is a quadratic equation. The Babylonian solution was verbal, as opposed to formulaic. Thus, in English it would look something like the following.

Divide the perimeter by 4.

Square the result of step 1.

Subtract the area from the result of step 2.

Find the square root of the result of step 3.

Add the result of step 4 to that of step 1 to obtain the base.

In modern notation the above translates as

, which is one of the two solutions that would be obtained using modern methods. Once

is solved for,

can be solved for using

or

. This ancient problem is still a standard textbook exercise for

algebra students

today.

In 628 AD, Brahmagupta published the general solution to the quadratic equation

which he gave verbally as, "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." This translates into modern notation as

which is one of the two solutions that would be obtained using modern methods. Finally, it was in 1637 that the French mathematician René Descartes published the solution of the quadratic equation

in its modern form:

Factoring

The method of solving quadratic equations by factoring rests on the principle of zero products, which states that if

and

are real numbers, then:

Another way to phrase this is that it is impossible to multiply two nonzero real numbers together and get a product of zero. The same principle applies when

and

are real-valued expressions, and this fact is exploited to solve quadratic equations by factoring.

This same basic program is used whenever solving quadratic equations by factoring. In this example the equation has integer solutions. This is not always the case, as in the following example, which has rational solutions.

The expression

is not always factorable using the methods of elementary algebra. In such cases equation

is best solved using other methods, as discussed below.

Taking the Square Root

This method for solving quadratic equations is based on the principle of square roots, which states that if

is a

real number

and

is a non-negative real number, then:

The same principle applies when

is a real-valued expression, and this can be used to solve quadratic equations by taking square roots.

The presence of the radical indicates that the solutions of the above equation are irrational. The factoring methods of elementary algebra are not able to turn up such solutions.

The presence of the imaginary unit

indicates that the solutions of the above equation are complex. Again, elementary factoring methods fail to reveal such solutions.

Completing the Square

The method of completing the square is based on an observation made on perfect square trinomials:

In both

and

it is observed that if one takes half of the first degree coefficient

and squares the result, one obtains the constant term of each polynomial

. This provides a procedure for obtaining a perfect square trinomial. For instance, in the expression

, the first degree coefficient is

. Based on the above observation, adding the constant

to the expression will result in a perfect square trinomial:

. The addition of

to

completes the perfect square trinomial

.

The method of solving quadratic equations by completing the square is based on the above in conjunction with the principle of square roots. It must be kept in mind that this method only works when the second degree coefficient is

.

The Quadratic Formula

The quadratic formula is based on the method of completing the square. This is explicitly shown by using that method to solve

, which is the most general quadratic equation. This is done below.

The quadratic formula reduces the solution of any quadratic equation to an arithmetic problem, as shown below.

The quantity

that appears under the radical in

is called the discriminant, and its value determines the nature of the solutions of

.

If

, then

has two distinct real solutions.

Example:

,

, Solutions:

.

If

, then

has a single real solution.

Example:

,

, Solution:

.

If

, then

has two non-real complex conjugate solutions.

Example:

,

, Solutions:

.

Bibliography

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Larson, Ron, and David C. Falvo. Precalculus. Boston: Cengage, 2014.

Lial, Margaret L., E. J. Hornsby, and Terry McGinnis. Intermediate Algebra. Boston: Pearson, 2012.

Miller, Julie. College Algebra Essentials. New York: McGraw, 2013.

Rudman, Peter S. How Mathematics Happened : The First 50,000 Years. Amherst, NY: Prometheus, 2007.

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