Before you read

Answer first — see what you already know.

A solar panel's area is $x^{2} - 81$ square meters. Which factored form represents the dimensions?

A
$(x - 9)^{2}$
B
$(x + 9) (x - 9)$
C
$(x + 81) (x - 1)$
D
Cannot be factored

Check your answer

Answer:

Before you read

Answer first — see what you already know.

The area of a cross-section is $9 x^{2} - 4$ cm $^{2}$ . What is its factored form?

A
$(9 x + 4) (x - 1)$
B
$(3 x + 2) (3 x - 2)$
C
$(3 x - 2)^{2}$
D
$(9 x + 2) (x - 2)$

Check your answer

Answer:

Theory

Three Special Factoring Patterns

Memorize these three patterns — they save enormous time on the SAT:

1. Difference of Squares:
$a^2 - b^2 = (a + b)(a - b)$

$x^2 - 16$
$(x+4)(x-4)$ difference of squares: a $^{2}$ -b $^{2}$ = (a+b)(a-b)

Difference of squares

2. Perfect Square Trinomial (sum):
$a^2 + 2ab + b^2 = (a + b)^2$

3. Perfect Square Trinomial (difference):
$a^2 - 2ab + b^2 = (a - b)^2$

How to recognize them:
- Difference of squares: two perfect squares separated by a minus sign, NO middle term
- Perfect square trinomial: first and last terms are perfect squares, middle term is $\pm 2 \times \sqrt{\text{first}} \times \sqrt{\text{last}}$

Theory

Difference of Squares — Deep Dive

This pattern appears constantly on the SAT.

Key: both terms must be perfect squares, connected by subtraction.

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

Example 1

Factor $x^2 - 25$

$x^2 = (x)^2$ and $25 = (5)^2$ .

$x^2 - 25 = (x+5)(x-5)$

(x+5)(x-5)

Example 2

Factor $4x^2 - 9$

$4x^2 = (2x)^2$ and $9 = (3)^2$ .

$4x^2 - 9 = (2x+3)(2x-3)$

(2x+3)(2x-3)

Example 3

Factor $16x^4 - 1$

$16x^4 = (4x^2)^2$ and $1 = (1)^2$ .

$16x^4 - 1 = (4x^2 + 1)(4x^2 - 1)$

But $4x^2 - 1$ is ALSO a difference of squares!

$= (4x^2 + 1)(2x + 1)(2x - 1)$

(4x^2 + 1)(2x + 1)(2x - 1)

Continue this lesson in the app

4 more sections including examples, practice problems, and step-by-step solutions.

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