Quadratic Factorization

Algebra

Factoriing Quadratics

Factorization of $A x^{2} + B x + C$ with $A = 1$

$x^{2} + (p + q) x + p q = (x + p) (x + q)$

Example 1

Factorize $x^{2} + 5 x + 6$

To factorise this expressions, find two numbers (p and q) that have a product of +6 and a sum of +5.

There are a couple of ways of making +6 by multiplying two numbers.

There are 1 × 6 and 2 × 3.

Only the combination of 2 and 3 will also give a sum of +5, so the two numbers are 2 and 3.

p = 2 and q = 3

$\begin{aligned} x^{2} + 5 x + 6 & = x^{2} + (2 + 3) x + (2 \times 3) \\ x^{2} + 5 x + 6 & = (x + 2) (x + 3) \end{aligned}$

Example 2

Factorize $x^{2} - 4 x - 12$

To factorise this expressions, find two numbers (p and q) that have a product of −12 and a sum of −4.

There are −(1 × 12), −(2 × 6), and −(3 × 4)

Only the combination of 2 and −6 will also give a sum of −4, so the two numbers are 2 and −6.

p = 2 and q = −6

$\begin{aligned} x^{2} - 4 x - 12 & = x^{2} + (2 - 6) x + (2) (- 6) \\ x^{2} - 4 x - 12 & = (x + 2) (x - 6) \end{aligned}$

Factorization of $A x^{2} + B x + C$ with $A \neq 1$

Example 1

Factorize $4 x^{2} + 23 x + 15$

$\begin{aligned} 4 x^{2} + 23 x + 15 & = 4 x^{2} + 20 x + 3 x + 15 \\ 4 x^{2} + 23 x + 15 & = 4 x (x + 5) + 3 (x + 5) \\ 4 x^{2} + 23 x + 15 & = (x + 5) (4 x + 3) \end{aligned}$

So, $4 x^{2} + 23 x + 15 = (x + 5) (4 x + 3)$

Example 2

Factorize $2 x^{2} + 11 x - 21$

$\begin{aligned} 2 x^{2} + 11 x - 21 & = 2 x^{2} + 14 x - 3 x - 21 \\ 2 x^{2} + 11 x - 21 & = 2 x (x + 7) - 3 (x + 7) \\ 2 x^{2} + 11 x - 21 & = (x + 7) (2 x - 3) \end{aligned}$

So, $2 x^{2} + 11 x - 21 = (x + 7) (2 x - 3)$

Exercise

--- Open this page ---

Factoriing Quadratics

Factorization of Ax2+Bx+C with A=1

Factorization of Ax2+Bx+C with A≠1

Exercise

Algebra

Factorization of $A x^{2} + B x + C$ with $A = 1$

Factorization of $A x^{2} + B x + C$ with $A \neq 1$