Basic Math Examples

$a^{4} + 2 a^{2} b^{2} + 9 b^{4}$

Step 1

Rewrite the middle term.

$a^{4} + 2 a^{2} (3 b^{2}) - 4 a^{2} b^{2} + 9 b^{4}$

Step 2

Rearrange terms.

$a^{4} + 2 a^{2} (3 b^{2}) + 9 b^{4} - 4 a^{2} b^{2}$

Step 3

Factor first three terms by perfect square rule.

${(a^{2} + 3 b^{2})}^{2} - 4 a^{2} b^{2}$

Step 4

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

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Multiply $2$ by $- 1$ .

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

Step 6.2

Remove parentheses.

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