Algebra Examples

$25 x^{2} + 16 - 40 x - 64 y^{2}$

Step 1

Factor using the perfect square rule.

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Step 1.1

Rearrange terms.

$25 x^{2} - 40 x + 16 - 64 y^{2}$

Step 1.2

Rewrite $25 x^{2}$ as ${(5 x)}^{2}$ .

${(5 x)}^{2} - 40 x + 16 - 64 y^{2}$

Step 1.3

Rewrite $16$ as $4^{2}$ .

${(5 x)}^{2} - 40 x + 4^{2} - 64 y^{2}$

Step 1.4

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

$40 x = 2 \cdot (5 x) \cdot 4$

Step 1.5

Rewrite the polynomial.

${(5 x)}^{2} - 2 \cdot (5 x) \cdot 4 + 4^{2} - 64 y^{2}$

Step 1.6

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 5 x$ and $b = 4$ .

${(5 x - 4)}^{2} - 64 y^{2}$

Step 2

Rewrite $64 y^{2}$ as ${(8 y)}^{2}$ .

${(5 x - 4)}^{2} - {(8 y)}^{2}$

Step 3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5 x - 4$ and $b = 8 y$ .

$(5 x - 4 + 8 y) (5 x - 4 - (8 y))$

Step 4

Multiply $8$ by $- 1$ .

$(5 x - 4 + 8 y) (5 x - 4 - 8 y)$