For all real values of x, the expression 9x^2 - 36 is equivalent to which of the following… — ACT

For all real values of $x$ , the expression $9x^2 - 36$ is equivalent to which of the following?

A
$(3x - 6)^2$
B
$36(x - 1)(x + 1)$
C
$(3x^2 - 6)(3x^2 + 6)$
$9(x - 2)(x + 2)$ Answer
E
$9(x - 2)^2$

The equivalent expression is

9(x - 2)(x + 2)

Factoring out the greatest common factor of

9

from the expression

9x^2 - 36

gives

9(x^2 - 4)

. The binomial

x^2 - 4

is a difference of squares (

x^2 - 2^2

), which can be factored into

(x - 2)(x + 2)

. Combining these parts results in the equivalent expression

9(x - 2)(x + 2)

Identify the greatest common factor (GCF) of the terms in the expression

9x^2 - 36

The GCF of

9x^2

and

36

9

Factoring out the GCF simplifies the remaining polynomial expression.

Factor out the GCF of

9

from the original expression.

9(x^2 - 4)

This separates the common numeric factor from the quadratic binomial.

Factor the remaining binomial expression

x^2 - 4

inside the parentheses.

x^2 - 4 = (x - 2)(x + 2)

The expression

x^2 - 4

is a difference of squares (

x^2 - 2^2

), which follows the factoring pattern

a^2 - b^2 = (a - b)(a + b)

Combine the factored components to write the final equivalent expression.

9(x - 2)(x + 2)

Combining the GCF with the factored binomial factors yields the completely factored equivalent expression.

Factoring a polynomial by first removing a greatest common factor and then applying the difference of squares formula.

Question