Difference of Squares: A Useful Formula

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The difference of squares is an expression of the form $a^{2} - b^{2}$ , where $a$ and $b$ can be numbers or variables.

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

x^{2} - 9 = x^{2} - 3^{2} = (x + 3) (x - 3)

Verification: $(x + 3) (x - 3) = x^{2} - 3 x + 3 x - 9 = x^{2} - 9$ ✓

Using the FOIL method:

(a + b) (a - b) = a^{2} - ab + ab - b^{2} = a^{2} - b^{2}

The terms $- ab$ and $+ ab$ cancel out!

25 - 16 = 5^{2} - 4^{2} = (5 + 4) (5 - 4) = 9 \cdot 1 = 9

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

4 x^{2} - 9 y^{2} = (2 x)^{2} - (3 y)^{2} = (2 x + 3 y) (2 x - 3 y)

2 x^{2} - 18 = 2 (x^{2} - 9) = 2 (x + 3) (x - 3)

5 3^{2} - 4 7^{2} = (53 + 47) (53 - 47) = 100 \cdot 6 = 600

(x + 1)^{2} - 4 = ((x + 1) + 2) ((x + 1) - 2) = (x + 3) (x - 1)

x^{2} + 9 \neq = (x + 3) (x + 3)

Sum of squares cannot be factored this way!

x^{2} - 7 \neq = (x + 7) (x - 7)

But this IS CORRECT! Remember that $7 \cdot 7 = 7$ .

Correct: $x^{2} - 7 = (x + 7) (x - 7)$

Practice difference of squares:

Formula	Form
Difference of squares	$a^{2} - b^{2} = (a + b) (a - b)$
Sum of squares	Cannot be factored (without complex numbers)
Perfect square	$(a + b)^{2} = a^{2} + 2 ab + b^{2}$