Factoring Quadratic Equations - GMAT Math Study Guide

Definitions

  • Quadratic Equation - An equation that can be written in the form ax2 + bx + c = 0.
    For example, 6x2 + 2x + 1 = 0 is a quadratic equation while 6x + 2 is not a quadratic equation.
  • Factoring - The process of breaking apart of an equation into factors (or separate terms) such that when the separate terms are multiplied together, they produce the original equation.
    For example, x2 - x - 2 = (x+1)(x-2). In this case, the equation x2 - x - 2 = 0 can be broken apart into two factors [i.e., (x+1)(x-2) = 0] such that when these two separate terms (i.e., factors) are multiplied together, the result is the original equation.

Basic Factoring

In factoring a basic quadratic equation such as x2 + 6x + 8 = 0, you must find two numbers that add to b (i.e., +6 in this case) and multiply to c (+8 in this case). The numbers +4 and +2 have the properties necessary. Consequently, (x + 4) and (x + 2) are the two factors.

To summarize:

In factoring a quadratic of the form x2 + bx + c, look for two numbers that add to b and multiply to c

Examples of Basic Factoring

If you have not done factoring in years or it is entirely new, you may be confused at this point. However, the following examples and explanation of going between factored and quadratic form should clarify most confusion.

x2 + x - 12 = 0
Find two numbers that add to +1 and multiply to -12.
Two such numbers are +4 and -3.
(x + 4)(x - 3) = 0
x = -4 or x = +3 since each value satisfies the equation (x + 4)(x - 3) = 0.

Another example:

x2 - 3x - 10 = 0
Find two numbers that add to -3 and multiply to -10.
Two such numbers are -5 and +2.
(x - 5)(x + 2) = 0
x = +5 or x = -2 since each value satisfies the equation (x - 5)(x + 2) = 0.

Another example:

x2 + 7x + 6 = 0
Find two numbers that add to +7 and multiply to +6.
Two such numbers are +6 and +1.
(x + 6)(x + 1) = 0
x = -6 or x = -1 since each value satisfies the equation (x + 6)(x + 1) = 0.

Reverse Factoring

The reverse of factoring is called FOIL, which stands for first, outer, inner, last. To acquire the quadratic form (ax2 + bx + c = 0) from the factored form [(x - a)(x - b) = 0]: (1) multiply the first terms, then the outer terms, then the inner terms, and finally the last terms (2) add each of the terms together and simplify. For example:

(x - 4)(x + 2) = ?
First: x(x) = x2
Outer: x(2) = 2x
Inner: (-4)(x) = -4x
Last: -4(2) = -8
(x - 4)(x + 2) = x2 + 2x - 4x - 8 = x2 - 2x - 8

Translating Between Factored and Quadratic Form

Factoring, as defined above, is the process of breaking apart of an equation into factors (or separate terms) such that when the separate terms are multiplied together, they produce the original equation. Factoring works on the following fundamental relationship:

(x - r1)(x - r2) = 0 = x2 + bx + c where r1 and r2 are the roots, or solutions, of the quadratic equation

As an example:

Roots: +6, -4
(x - 6)(x + 4) = ?
x2 + 4x - 6x - 24
x2 - 2x - 24

Consequently, if you saw x2 - 2x - 24 = 0 as a question, you could quickly solve it by factoring it as follows:

x2 - 2x - 24 = 0
(x - 6)(x + 4) = 0
x = 6, -4 since both of these values make the equation (x - 6)(x + 4) = 0 true.

Three Common Forms

There are three common forms that are easily factored. It is essential that you can recognize these three factored forms and quickly work with them:

Difference of Squares

a2 - b2 = (a + b)(a - b)

For example:

x2 - 4 = (x + 2)(x - 2)
a = x, b = +2

A Plus B Squared

a2 + 2ab + b2 = (a + b)2

For example:

x2 + 4x + 4 = (x + 2)2
a = x, b = +2

A Minus B Squared

a2 - 2ab + b2 = (a - b)2

For example:

x2 - 4x + 4 = (x - 2)2
a = x, b = +2

Dividing By Zero: Undefined

The rules of mathematics and division in particular state that you cannot divide by zero. Consequently, x divided by zero is undefined just as 1 divided by zero is undefined and 0 divided by 0 is undefined. Further, if you are factoring an equation with a variable in the denominator, any value of that variable that makes the denominator zero is not a legitimate solution. This is best explained and understood with examples.


Begin by factoring the equation as much as possible.
Top: Two numbers that add to +4 and multiply to -12 are +6 and -2.
Bottom: Two numbers that add to +5 and multiply to +4 are +1 and +4.

The solutions (or roots) are x = -6 or x = +2. Since the denominator cannot equal zero, x = -4 or x = -1 [both values that cause the entire denominator to be equal to zero] are not solutions but are instead values of x that cause the entire expression to be undefined.

Another Example


Begin by factoring the equation as much as possible.
Top: Two numbers that add to -10 and multiply to +21 are -7 and -3.
Bottom: Two numbers that add to -6 and multiply to +8 are -4 and -2.

The solutions (or roots) are x = +3 or x = +7. Since the denominator cannot equal zero, x = +2 or x = +4 [both values that cause the entire denominator to be equal to zero] are not solutions but are instead values of x that cause the entire expression to be undefined.

Solutions Summary

Quadratic equations can have zero, one, or two real solutions.

x2 + 9 = 0; No real solution
x2 + 6x + 9 = 0; One real solution: x = -3
x2 - 4 = 0; Two real solutions: x = 2 or x = -2