Algebra Examples

$x^{2} - a^{2} - 2 a b - b^{2}$

Step 1

Regroup terms.

$x^{2} - a^{2} - 2 a b - b^{2}$

Step 2

Factor $- 1$ out of $- a^{2} - 2 a b - b^{2}$ .

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Step 2.1

Factor $- 1$ out of $- a^{2}$ .

$x^{2} - (a^{2}) - 2 a b - b^{2}$

Step 2.2

Factor $- 1$ out of $- 2 a b$ .

$x^{2} - (a^{2}) - (2 a b) - b^{2}$

Step 2.3

Factor $- 1$ out of $- b^{2}$ .

$x^{2} - (a^{2}) - (2 a b) - (b^{2})$

Step 2.4

Factor $- 1$ out of $- (a^{2}) - (2 a b)$ .

$x^{2} - (a^{2} + 2 a b) - (b^{2})$

Step 2.5

Factor $- 1$ out of $- (a^{2} + 2 a b) - (b^{2})$ .

$x^{2} - (a^{2} + 2 a b + b^{2})$

Step 3

Factor using the perfect square rule.

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Step 3.1

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 a b = 2 \cdot a \cdot b$

Step 3.2

Rewrite the polynomial.

$x^{2} - (a^{2} + 2 \cdot a \cdot b + b^{2})$

Step 3.3

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = a$ and $b = b$ .

$x^{2} - {(a + b)}^{2}$

Step 4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = a + b$ .

$(x + a + b) (x - (a + b))$

Step 5

Apply the distributive property.

$(x + a + b) (x - a - b)$