Sitemap# Factoring by Completing the Square

Top of PageAbout UsPrivacy

Mobile Math Website

All quadratic equations in one variable can be solved using the method of Completing the Square. This method depends upon the fact:

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

The middle term is twice the product of the square roots of the other two terms:

2ab = 2√a2√b2

Factor 5x2 − 30 = x2 − 5

5x2 − x2 − 30 + 5 = 0

4x2 − 25 = 0

x2 = 25/4

x = ±√(25/4) = ±5/2

Factor the difficult equation,

2x2 − 7x = x + 14

Place all math terms on one side of the equation.

2x2 − x − 7x − 14 = 0

2x2 − 8x − 14 = 0

x2 − 4x − 7 = 0

There is no number combination of the constant 14, [ 1, 14 ] and [ 2, 7 ], that when factored using the “(x )(x )” setup are equivalent to the original equation.

However, x2 − 4x of the quadratic equation is almost a perfect square. The perfect trinomial square x2 − 4x + 4 has factors:

(x − 2) (x − 2).

(x − 2) (x − 2) = x2 − 4x + 4

Isolate the almost perfect square to one side of the equation (very important math step).

x2 − 4x − 7 = 0

x2 − 4x = 7

Use the math Substitution Property.

x2 − 4x = 7

(x − 2) (x − 2) = 7 + 4

(x − 2)2 = 11

x − 2 = ±√11

x = 2 ±√11

To determine the missing term of x2 + 6x to complete a perfect trinomial square:

Apply the form (a + b)2 = a2 + 2ab + b2,

x2 + 6x = a2 + 6a, and

2ab = 6a, then

2b = 6

b = 3, and

(a + 3)2 = a2 + 2a3 + 32, and

(a + 3) (a + 3) = a2 + 6a + 9, when a = 3

Check the equation is correct for values of a = 3 and b = 3

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

(3 + 3)2 = 32 + (2) (3) (3) + 32

62 = 9 + 18 + 9

36 = 36 is correct.

The perfect trinomial math square is:

x2 + 6x + 9 = 36, can be factored as

(x + 3)2 = (3 + 3) (3 + 3) = 36 when x = 3,

and equals 0 when x = −3

Copyright © DigitMath.com

All Rights Reserved.