Algebra Examples

$(\frac{a^{2} - 4 a b + 4 b^{2}}{a^{2} + 2 a b}) (\frac{a^{2}}{a^{2} - 4 b^{2}})$

Step 1

Factor using the perfect square rule.

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Step 1.1

Rewrite $4 b^{2}$ as ${(2 b)}^{2}$ .

$\frac{a^{2} - 4 a b + {(2 b)}^{2}}{a^{2} + 2 a b} \cdot \frac{a^{2}}{a^{2} - 4 b^{2}}$

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.

$4 a b = 2 \cdot a \cdot (2 b)$

Step 1.3

Rewrite the polynomial.

$\frac{a^{2} - 2 \cdot a \cdot (2 b) + {(2 b)}^{2}}{a^{2} + 2 a b} \cdot \frac{a^{2}}{a^{2} - 4 b^{2}}$

Step 1.4

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

$\frac{{(a - 2 b)}^{2}}{a^{2} + 2 a b} \cdot \frac{a^{2}}{a^{2} - 4 b^{2}}$

Step 2

Factor $a$ out of $a^{2} + 2 a b$ .

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Step 2.1

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

$\frac{{(a - 2 b)}^{2}}{a \cdot a + 2 a b} \cdot \frac{a^{2}}{a^{2} - 4 b^{2}}$

Step 2.2

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

$\frac{{(a - 2 b)}^{2}}{a \cdot a + a (2 b)} \cdot \frac{a^{2}}{a^{2} - 4 b^{2}}$

Step 2.3

Factor $a$ out of $a \cdot a + a (2 b)$ .

$\frac{{(a - 2 b)}^{2}}{a (a + 2 b)} \cdot \frac{a^{2}}{a^{2} - 4 b^{2}}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $4 b^{2}$ as ${(2 b)}^{2}$ .

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

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 $b = 2 b$ .

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

Step 3.3

Multiply $2$ by $- 1$ .

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

Step 4

Simplify terms.

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Step 4.1

Combine.

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

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

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

Step 4.2.2.2

Cancel the common factor.

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

Step 4.2.2.3

Rewrite the expression.

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

Step 4.3

Cancel the common factor of $a^{2}$ and $a$ .

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Step 4.3.1

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

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

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

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

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

Step 4.3.2.2

Cancel the common factor.

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

Step 4.3.2.3

Rewrite the expression.

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

Step 5

Simplify the denominator.

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Step 5.1

Raise $a + 2 b$ to the power of $1$ .

$\frac{(a - 2 b) a}{{(a + 2 b)}^{1} (a + 2 b)}$

Step 5.2

Raise $a + 2 b$ to the power of $1$ .

$\frac{(a - 2 b) a}{{(a + 2 b)}^{1} {(a + 2 b)}^{1}}$

Step 5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(a - 2 b) a}{{(a + 2 b)}^{1 + 1}}$

Step 5.4

Add $1$ and $1$ .

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

Step 6

Reorder factors in $\frac{(a - 2 b) a}{{(a + 2 b)}^{2}}$ .

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