4.4 Solve ax² + bx + c = 0 by Factoring

Example 1

Factor ax² + bx + c where c > 0

Factor 2x² + 9x + 7.

You want 2x² + 9x + 7 = (kx + m)(lx + n) where k and l are factors of _2_ and m and n are factors of _7_. Because mn _>_ 0, m and n have the same sign. So, m and n must both be _positive_ because the coefficient of x, 9, is _positive_.

k, l	m, n	(kx + m)(lx + n)	ax^2 + bx + c
2, 1	1, _7_	(_2x + 1_)(_x + 7_)	_2x^2 + 15x + 7_
2, 1	7, _1_	(_2x + 7_)(_x + 1_)	_2x^2 + 9x+ 7_

The correct factorization is

2x² + 9x + 7 = (_2x + 7_)(_x + 1_).

Example 2

Factor ax² + bx + c where c < 0

Factor 3x² - x - 2.

You want 3x² - x - 2 = (kx + m)(lx + n) where k and l are factors of _3_ and m and n are factors of _-2_. Because mn _<_ 0, m and n have the _opposite_ signs.

k, l	m, n	(kx + m)(lx + n)	ax^2 + bx + c
3, 1	-1, _2_	(_3x - 1_)(_x + 2_)	_3x^2 + 5x - 2_
3, 1	2, _-1_	(_3x + 2_)(_x - 1_)	_3x^2 - x - 2_
3, 1	l, _-2_	(_3x - 1_)(_x + 2_)	_3x^2 - 5x - 2_
3, 1	-2, _1_	(_3x + 2_)(_x - 1_)	_3x^2 + x - 2_

The correct factorization is

3x² - x - 2 = (_3x + 2_)(_x - 1_).

Example 3

Factor with special patterns

Factor the expression.

a. 16x^2 - 36 = (_ 4x_)^2 - _6_^2 = (_4x + 6_)(_4x - 6_)	Difference of two squares
b. 9y^2 + 42y + 49 = (_3y_)^2 + 2(_3y_)(_7_) + _7_^2 = (_3y + 7_)^2	Perfect square trinomial
c. 25t^2 - 110t + 121 = (_5t_)^2 - 2(_5t_)(_11_) + _11_^2 = (_5t - 11_)^2	Perfect square trinomial

Example 4

Factor the expression.

a. 4x² - 4 = 4(_x² - 1_) = 4(_x + 1_)(_x - 1_)

b. -3y² - 18y = -3y(_y + 6_)

c. -4m² - 10m + 24 = -2(_2m² + 5m - 12_)

= -2(_2m - 3_)(_m + 4_)

d. 5z² - 25z + 40 = 5(_z² - 5z + 8_)

Example 5

Solve quadratic equations

a. 2x^2 - x - 21 = 0	Original equation
(_2x - 7_)(_x + 3_) = 0	Factor.
_2x - 7_ = 0 or _x + 3_ = 0	Zero product property
x =___ or x = _-3_	Solve for x.
b. 4r^2 - 18r + 24 = 6r -12	Original equation
_4r^2 - 24r + 36_ = 0	Standard form
_r^2 - 6r + 9_ = 0	Divide each side by _4_.
_(r - 3)^2_ = 0	Factor.
_r - 3_ = 0	Zero product property
r = _3_	Solve for r.