Algebra Examples

$\frac{4 x^{2} - 1}{5 x - 10 x^{2}}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{{(2 x)}^{2} - 1}{5 x - 10 x^{2}}$

Step 1.2

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

$\frac{{(2 x)}^{2} - 1^{2}}{5 x - 10 x^{2}}$

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 $b = 1$ .

$\frac{(2 x + 1) (2 x - 1)}{5 x - 10 x^{2}}$

Step 2

Simplify terms.

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Step 2.1

Factor $5 x$ out of $5 x - 10 x^{2}$ .

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Step 2.1.1

Factor $5 x$ out of $5 x$ .

$\frac{(2 x + 1) (2 x - 1)}{5 x (1) - 10 x^{2}}$

Step 2.1.2

Factor $5 x$ out of $- 10 x^{2}$ .

$\frac{(2 x + 1) (2 x - 1)}{5 x (1) + 5 x (- 2 x)}$

Step 2.1.3

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

$\frac{(2 x + 1) (2 x - 1)}{5 x (1 - 2 x)}$

Step 2.2

Cancel the common factor of $2 x - 1$ and $1 - 2 x$ .

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Step 2.2.1

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

$\frac{(2 x + 1) (- (- 2 x) - 1)}{5 x (1 - 2 x)}$

Step 2.2.2

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

$\frac{(2 x + 1) (- (- 2 x) - 1 (1))}{5 x (1 - 2 x)}$

Step 2.2.3

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

$\frac{(2 x + 1) (- (- 2 x + 1))}{5 x (1 - 2 x)}$

Step 2.2.4

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

$\frac{(2 x + 1) (- 1 (- 2 x + 1))}{5 x (1 - 2 x)}$

Step 2.2.5

Reorder terms.

$\frac{(2 x + 1) (- 1 (- 2 x + 1))}{5 x (- 2 x + 1)}$

Step 2.2.6

Cancel the common factor.

$\frac{(2 x + 1) (- 1 (- 2 x + 1))}{5 x (- 2 x + 1)}$

Step 2.2.7

Rewrite the expression.

$\frac{(2 x + 1) \cdot (- 1)}{5 x}$

Step 2.3

Simplify the expression.

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Step 2.3.1

Move $- 1$ to the left of $2 x + 1$ .

$\frac{- 1 \cdot (2 x + 1)}{5 x}$

Step 2.3.2

Move the negative in front of the fraction.

$- \frac{2 x + 1}{5 x}$