Algebra Examples

$\frac{4 {(- x)}^{2} - 1}{{(- x)}^{2}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $4 {(- x)}^{2}$ as ${(2 (- x))}^{2}$ .

$\frac{{(2 (- x))}^{2} - 1}{{(- x)}^{2}}$

Step 1.2

Rewrite $1$ as $1^{2}$ .

$\frac{{(2 (- x))}^{2} - 1^{2}}{{(- x)}^{2}}$

Step 1.3

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

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

Step 1.4

Simplify.

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Step 1.4.1

Multiply $- 1$ by $2$ .

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

Step 1.4.2

Multiply $- 1$ by $2$ .

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

Step 2

Simplify the denominator.

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Step 2.1

Apply the product rule to $- x$ .

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

Step 2.2

Raise $- 1$ to the power of $2$ .

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

Step 2.3

Multiply $x^{2}$ by $1$ .

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

Step 3

Simplify with factoring out.

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Step 3.1

Factor $- 1$ out of $- 2 x$ .

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

Step 3.2

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

$\frac{(- (2 x) - 1 (- 1)) (- 2 x - 1)}{x^{2}}$

Step 3.3

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

$\frac{- (2 x - 1) (- 2 x - 1)}{x^{2}}$

Step 3.4

Rewrite $- (2 x - 1)$ as $- 1 (2 x - 1)$ .

$\frac{- 1 (2 x - 1) (- 2 x - 1)}{x^{2}}$

Step 3.5

Factor $- 1$ out of $- 2 x$ .

$\frac{- 1 (2 x - 1) (- (2 x) - 1)}{x^{2}}$

Step 3.6

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

$\frac{- 1 (2 x - 1) (- (2 x) - 1 (1))}{x^{2}}$

Step 3.7

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

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

Step 3.8

Simplify the expression.

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Step 3.8.1

Rewrite $- (2 x + 1)$ as $- 1 (2 x + 1)$ .

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

Step 3.8.2

Multiply $- 1$ by $- 1$ .

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

Step 3.8.3

Multiply $2 x - 1$ by $1$ .

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