Let's review how to factor a expression of the form $x^{2} + b x + c$ first, then we will move to one of the form $a x^{2} + b x + c$ when $a > 1 .$

Factoring Expressions of the Form $x^{2} + b x + c$

Let's factor the expression

x^{2} + 2 x - 24 .

x^{2} + b x + c x^{2} + 2 x - 24

To factor this expression we need to find the factors of

c

such that their sum is

b .

For this case, we need numbers whose

product

is

24

and

sum

is

2 .

Since

- 24 < 0,

the factors should have different signs. And, since

2 > 0,

the greatest factor should be positive. Let's check these requirements for different factors of

- 24 .

Factors of $- 24$	Sum
$- 1, 24$	$- 1 + 24 = 23 \times$
$- 2, 12$	$- 2 + 12 = 10 \times$
$- 3, 8$	$- 3 + 8 = 5 \times$
$- 4, 6$	$- 4 + 6 = 2 ✓$

With this in mind, we can factor our original expression.

x^{2} + 2 x - 24 = 3 (x - 4) (x + 6)

Factoring Expressions of the Form $a x^{2} + b x + c$ When $a > 1$

For expressions of this form it is recommended to start by looking for the greatest common factor (GCF). This is the common factor with the greatest coefficient and the greatest exponent. Let's consider the case

3 x^{2} + 6 x - 72,

for instance. In this case the GCF is

3 .

3 x^{2} + 6 x - 72 3 x^{2} + 6 x - 72 = 3 (x^{2}) + 3 (2 x) - 3 (24) = 3 (x^{2} + 2 x - 24)

We can see that the expression inside the parentheses is the same from the previous case. However, to reduce a expression of the more general form

a x^{2} + b x + c

to one of the form

x^{2} + b x + c

is not always possible. If we consider

2 x^{2} + 11 x + 12,

for instance, the GCF is

1 .

In that case we have to proceed differently.

a x^{2} + b x + c 2 x^{2} + 11 x + 12

We need to look for factors of

a c

such that their sum is

b .

In this case,

a c = (2) (12) = 24 .

Since

24 > 0

we will need the factors to have the same sign. And, since

11 > 0,

we know both should be positive. Let's check for these requirements for different factors of

24 .

Factors of $24$	Sum
$1, 24$	$1 + 12 = 11 \times$
$2, 12$	$2 + 6 = 4 \times$
$3, 8$	$3 + 8 = 11 ✓$
$4, 6$	$3 + 4 = \times$

Now we rewrite our original expression using the factors that met the requirements mentioned before.

2 x^{2} + 11 x + 12 = 2 x^{2} + 3 x + 8 x + 12

We continue by looking for the GCF of the first two terms and the two last terms.

2 x^{2} + 3 x + 8 x + 12 = x (2 x + 3) + 4 (2 x + 3)

Finally, notice that both expressions have as a common factor the binomial

(2 x + 3) .

We can factor the expression once more.

x (2 x + 3) + 4 (2 x + 3) = (2 x + 3) (x + 4)

We found that the expression

2 x^{2} + 11 x + 12

can be factored as

(2 x + 3) (x + 4) .

Factoring Expressions of the Form x2+bx+c

Factoring Expressions of the Form ax2+bx+c When a>1

Factoring Expressions of the Form $x^{2} + b x + c$

Factoring Expressions of the Form $a x^{2} + b x + c$ When $a > 1$

{{ 'ml-heading-exercise' | message }} {{ exercise.name }} {{ 'ml-label-page' | message }} {{ exercise.page }}

{{ 'ml-solutions-missing-heading' | message }}