Algebra Examples

$\frac{2 a b}{a^{2} - 6 a b + 9 b^{2}} \cdot (a^{2} - 9 b^{2})$

Step 1

Factor using the perfect square rule.

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Step 1.1

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

$\frac{2 a b}{a^{2} - 6 a b + {(3 b)}^{2}} \cdot (a^{2} - 9 b^{2})$

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.

$6 a b = 2 \cdot a \cdot (3 b)$

Step 1.3

Rewrite the polynomial.

$\frac{2 a b}{a^{2} - 2 \cdot a \cdot (3 b) + {(3 b)}^{2}} \cdot (a^{2} - 9 b^{2})$

Step 1.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = a$ and $b = 3 b$ .

$\frac{2 a b}{{(a - 3 b)}^{2}} \cdot (a^{2} - 9 b^{2})$

Step 2

Multiply $\frac{2 a b}{{(a - 3 b)}^{2}}$ by $a^{2} - 9 b^{2}$ .

$\frac{2 a b (a^{2} - 9 b^{2})}{{(a - 3 b)}^{2}}$

Step 3

Simplify the numerator.

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Step 3.1

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

$\frac{2 a b (a^{2} - {(3 b)}^{2})}{{(a - 3 b)}^{2}}$

Step 3.2

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

$\frac{2 a b ((a + 3 b) (a - (3 b)))}{{(a - 3 b)}^{2}}$

Step 3.3

Multiply $3$ by $- 1$ .

$\frac{2 a b (a + 3 b) (a - 3 b)}{{(a - 3 b)}^{2}}$

Step 4

Cancel the common factor of $a - 3 b$ and ${(a - 3 b)}^{2}$ .

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Step 4.1

Factor $a - 3 b$ out of $2 a b (a + 3 b) (a - 3 b)$ .

$\frac{(a - 3 b) (2 a b (a + 3 b))}{{(a - 3 b)}^{2}}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $a - 3 b$ out of ${(a - 3 b)}^{2}$ .

$\frac{(a - 3 b) (2 a b (a + 3 b))}{(a - 3 b) (a - 3 b)}$

Step 4.2.2

Cancel the common factor.

$\frac{(a - 3 b) (2 a b (a + 3 b))}{(a - 3 b) (a - 3 b)}$

Step 4.2.3

Rewrite the expression.

$\frac{2 a b (a + 3 b)}{a - 3 b}$