Algebra Examples

$\frac{x}{25 - x^{2}} + \frac{2}{3 x - 15}$

Step 1

Simplify each term.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

Rewrite $25$ as $5^{2}$ .

$\frac{x}{5^{2} - x^{2}} + \frac{2}{3 x - 15}$

Step 1.1.2

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

$\frac{x}{(5 + x) (5 - x)} + \frac{2}{3 x - 15}$

Step 1.2

Factor $3$ out of $3 x - 15$ .

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Step 1.2.1

Factor $3$ out of $3 x$ .

$\frac{x}{(5 + x) (5 - x)} + \frac{2}{3 (x) - 15}$

Step 1.2.2

Factor $3$ out of $- 15$ .

$\frac{x}{(5 + x) (5 - x)} + \frac{2}{3 x + 3 \cdot - 5}$

Step 1.2.3

Factor $3$ out of $3 x + 3 \cdot - 5$ .

$\frac{x}{(5 + x) (5 - x)} + \frac{2}{3 (x - 5)}$

Step 2

Simplify with factoring out.

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Step 2.1

Factor $- 1$ out of $x$ .

$\frac{x}{(5 + x) (5 - x)} + \frac{2}{3 (- 1 (- x) - 5)}$

Step 2.2

Rewrite $- 5$ as $- 1 (5)$ .

$\frac{x}{(5 + x) (5 - x)} + \frac{2}{3 (- 1 (- x) - 1 (5))}$

Step 2.3

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

$\frac{x}{(5 + x) (5 - x)} + \frac{2}{3 (- 1 (- x + 5))}$

Step 2.4

Simplify the expression.

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Step 2.4.1

Move a negative from the denominator of $\frac{2}{3 (- 1 (- x + 5))}$ to the numerator.

$\frac{x}{(5 + x) (5 - x)} + \frac{- 1 \cdot 2}{3 (- x + 5)}$

Step 2.4.2

Reorder terms.

$\frac{x}{(5 + x) (- x + 5)} + \frac{- 1 \cdot 2}{3 (- x + 5)}$

Step 3

To write $\frac{x}{(5 + x) (- x + 5)}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{x}{(5 + x) (- x + 5)} \cdot \frac{3}{3} + \frac{- 1 \cdot 2}{3 (- x + 5)}$

Step 4

To write $\frac{- 1 \cdot 2}{3 (- x + 5)}$ as a fraction with a common denominator, multiply by $\frac{5 + x}{5 + x}$ .

$\frac{x}{(5 + x) (- x + 5)} \cdot \frac{3}{3} + \frac{- 1 \cdot 2}{3 (- x + 5)} \cdot \frac{5 + x}{5 + x}$

Step 5

Write each expression with a common denominator of $(5 + x) (- x + 5) \cdot 3$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.1

Multiply $\frac{x}{(5 + x) (- x + 5)}$ by $\frac{3}{3}$ .

$\frac{x \cdot 3}{(5 + x) (- x + 5) \cdot 3} + \frac{- 1 \cdot 2}{3 (- x + 5)} \cdot \frac{5 + x}{5 + x}$

Step 5.2

Multiply $\frac{- 1 \cdot 2}{3 (- x + 5)}$ by $\frac{5 + x}{5 + x}$ .

$\frac{x \cdot 3}{(5 + x) (- x + 5) \cdot 3} + \frac{- 1 \cdot 2 (5 + x)}{3 (- x + 5) (5 + x)}$

Step 5.3

Reorder the factors of $(5 + x) (- x + 5) \cdot 3$ .

$\frac{x \cdot 3}{3 (- x + 5) (5 + x)} + \frac{- 1 \cdot 2 (5 + x)}{3 (- x + 5) (5 + x)}$

Step 6

Combine the numerators over the common denominator.

$\frac{x \cdot 3 - 1 \cdot 2 (5 + x)}{3 (- x + 5) (5 + x)}$

Step 7

Simplify the numerator.

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Step 7.1

Move $3$ to the left of $x$ .

$\frac{3 \cdot x - 1 \cdot 2 (5 + x)}{3 (- x + 5) (5 + x)}$

Step 7.2

Multiply $- 1$ by $2$ .

$\frac{3 x - 2 (5 + x)}{3 (- x + 5) (5 + x)}$

Step 7.3

Apply the distributive property.

$\frac{3 x - 2 \cdot 5 - 2 x}{3 (- x + 5) (5 + x)}$

Step 7.4

Multiply $- 2$ by $5$ .

$\frac{3 x - 10 - 2 x}{3 (- x + 5) (5 + x)}$

Step 7.5

Subtract $2 x$ from $3 x$ .

$\frac{x - 10}{3 (- x + 5) (5 + x)}$

Step 8

Simplify with factoring out.

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Step 8.1

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

$\frac{x - 10}{3 (- (x) + 5) (5 + x)}$

Step 8.2

Rewrite $5$ as $- 1 (- 5)$ .

$\frac{x - 10}{3 (- (x) - 1 (- 5)) (5 + x)}$

Step 8.3

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

$\frac{x - 10}{3 (- (x - 5)) (5 + x)}$

Step 8.4

Rewrite negatives.

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Step 8.4.1

Rewrite $- (x - 5)$ as $- 1 (x - 5)$ .

$\frac{x - 10}{3 (- 1 (x - 5)) (5 + x)}$

Step 8.4.2

Move the negative in front of the fraction.

$- \frac{x - 10}{(3 (x - 5)) (5 + x)}$

Step 8.4.3

Reorder factors in $- \frac{x - 10}{(3 (x - 5)) (5 + x)}$ .

$- \frac{x - 10}{3 (x - 5) (5 + x)}$