Algebra Examples

${(x - 4)}^{2} - {(9 x - 1)}^{2}$

Step 1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x - 4$ and $b = 9 x - 1$ .

$(x - 4 + 9 x - 1) (x - 4 - (9 x - 1))$

Step 2

Simplify.

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Step 2.1

Add $x$ and $9 x$ .

$(10 x - 4 - 1) (x - 4 - (9 x - 1))$

Step 2.2

Subtract $1$ from $- 4$ .

$(10 x - 5) (x - 4 - (9 x - 1))$

Step 2.3

Factor $5$ out of $10 x - 5$ .

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Step 2.3.1

Factor $5$ out of $10 x$ .

$(5 (2 x) - 5) (x - 4 - (9 x - 1))$

Step 2.3.2

Factor $5$ out of $- 5$ .

$(5 (2 x) + 5 (- 1)) (x - 4 - (9 x - 1))$

Step 2.3.3

Factor $5$ out of $5 (2 x) + 5 (- 1)$ .

$5 (2 x - 1) (x - 4 - (9 x - 1))$

Step 2.4

Apply the distributive property.

$5 (2 x - 1) (x - 4 - (9 x) - - 1)$

Step 2.5

Multiply $9$ by $- 1$ .

$5 (2 x - 1) (x - 4 - 9 x - - 1)$

Step 2.6

Multiply $- 1$ by $- 1$ .

$5 (2 x - 1) (x - 4 - 9 x + 1)$

Step 2.7

Subtract $9 x$ from $x$ .

$5 (2 x - 1) (- 8 x - 4 + 1)$

Step 2.8

Add $- 4$ and $1$ .

$5 (2 x - 1) (- 8 x - 3)$

Step 3

Factor.

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Step 3.1

Factor $- 1$ out of $- 8 x - 3$ .

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Step 3.1.1

Factor $- 1$ out of $- 8 x$ .

$5 (2 x - 1) (- (8 x) - 3)$

Step 3.1.2

Rewrite $- 3$ as $- 1 (3)$ .

$5 (2 x - 1) (- (8 x) - 1 (3))$

Step 3.1.3

Factor $- 1$ out of $- (8 x) - 1 (3)$ .

$5 (2 x - 1) (- (8 x + 3))$

Step 3.2

Remove unnecessary parentheses.

$5 (2 x - 1) \cdot - 1 (8 x + 3)$

Step 4

Combine exponents.

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Step 4.1

Factor out negative.

$- (5 (2 x - 1) (8 x + 3))$

Step 4.2

Multiply $5$ by $- 1$ .

$- 5 ((2 x - 1) (8 x + 3))$

Step 5

Remove unnecessary parentheses.

$- 5 (2 x - 1) (8 x + 3)$