Calculus Examples

$x^{2} + 12 x + 36 - 9 y^{2}$

Step 1

Factor using the perfect square rule.

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Step 1.1

Rewrite $36$ as $6^{2}$ .

$x^{2} + 12 x + 6^{2} - 9 y^{2}$

Step 1.2

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

$12 x = 2 \cdot x \cdot 6$

Step 1.3

Rewrite the polynomial.

$x^{2} + 2 \cdot x \cdot 6 + 6^{2} - 9 y^{2}$

Step 1.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 6$ .

${(x + 6)}^{2} - 9 y^{2}$

Step 2

Rewrite $9 y^{2}$ as ${(3 y)}^{2}$ .

${(x + 6)}^{2} - {(3 y)}^{2}$

Step 3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x + 6$ and $b = 3 y$ .

$(x + 6 + 3 y) (x + 6 - (3 y))$

Step 4

Multiply $3$ by $- 1$ .

$(x + 6 + 3 y) (x + 6 - 3 y)$