Algebra Examples

$\frac{2 x}{x + 5} - \frac{10}{x - 5} = \frac{2 x^{2} + 50}{x^{2} - 25}$

Step 1

Subtract $\frac{2 x^{2} + 50}{x^{2} - 25}$ from both sides of the equation.

$\frac{2 x}{x + 5} - \frac{10}{x - 5} - \frac{2 x^{2} + 50}{x^{2} - 25} = 0$

Step 2

Simplify $\frac{2 x}{x + 5} - \frac{10}{x - 5} - \frac{2 x^{2} + 50}{x^{2} - 25}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Factor $2$ out of $2 x^{2} + 50$ .

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Step 2.1.1.1

Factor $2$ out of $2 x^{2}$ .

$\frac{2 x}{x + 5} - \frac{10}{x - 5} - \frac{2 (x^{2}) + 50}{x^{2} - 25} = 0$

Step 2.1.1.2

Factor $2$ out of $50$ .

$\frac{2 x}{x + 5} - \frac{10}{x - 5} - \frac{2 x^{2} + 2 \cdot 25}{x^{2} - 25} = 0$

Step 2.1.1.3

Factor $2$ out of $2 x^{2} + 2 \cdot 25$ .

$\frac{2 x}{x + 5} - \frac{10}{x - 5} - \frac{2 (x^{2} + 25)}{x^{2} - 25} = 0$

Step 2.1.2

Simplify the denominator.

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Step 2.1.2.1

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

$\frac{2 x}{x + 5} - \frac{10}{x - 5} - \frac{2 (x^{2} + 25)}{x^{2} - 5^{2}} = 0$

Step 2.1.2.2

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

$\frac{2 x}{x + 5} - \frac{10}{x - 5} - \frac{2 (x^{2} + 25)}{(x + 5) (x - 5)} = 0$

Step 2.2

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

$\frac{2 x}{x + 5} \cdot \frac{x - 5}{x - 5} - \frac{10}{x - 5} - \frac{2 (x^{2} + 25)}{(x + 5) (x - 5)} = 0$

Step 2.3

To write $- \frac{10}{x - 5}$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

$\frac{2 x}{x + 5} \cdot \frac{x - 5}{x - 5} - \frac{10}{x - 5} \cdot \frac{x + 5}{x + 5} - \frac{2 (x^{2} + 25)}{(x + 5) (x - 5)} = 0$

Step 2.4

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

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Step 2.4.1

Multiply $\frac{2 x}{x + 5}$ by $\frac{x - 5}{x - 5}$ .

$\frac{2 x (x - 5)}{(x + 5) (x - 5)} - \frac{10}{x - 5} \cdot \frac{x + 5}{x + 5} - \frac{2 (x^{2} + 25)}{(x + 5) (x - 5)} = 0$

Step 2.4.2

Multiply $\frac{10}{x - 5}$ by $\frac{x + 5}{x + 5}$ .

$\frac{2 x (x - 5)}{(x + 5) (x - 5)} - \frac{10 (x + 5)}{(x - 5) (x + 5)} - \frac{2 (x^{2} + 25)}{(x + 5) (x - 5)} = 0$

Step 2.4.3

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

$\frac{2 x (x - 5)}{(x + 5) (x - 5)} - \frac{10 (x + 5)}{(x + 5) (x - 5)} - \frac{2 (x^{2} + 25)}{(x + 5) (x - 5)} = 0$

Step 2.5

Combine the numerators over the common denominator.

$\frac{2 x (x - 5) - 10 (x + 5)}{(x + 5) (x - 5)} - \frac{2 (x^{2} + 25)}{(x + 5) (x - 5)} = 0$

Step 2.6

Combine the numerators over the common denominator.

$\frac{2 x (x - 5) - 10 (x + 5) - 2 (x^{2} + 25)}{(x + 5) (x - 5)} = 0$

Step 2.7

Simplify each term.

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Step 2.7.1

Apply the distributive property.

$\frac{2 x \cdot x + 2 x \cdot - 5 - 10 (x + 5) - 2 (x^{2} + 25)}{(x + 5) (x - 5)} = 0$

Step 2.7.2

Multiply $x$ by $x$ by adding the exponents.

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Step 2.7.2.1

Move $x$ .

$\frac{2 (x \cdot x) + 2 x \cdot - 5 - 10 (x + 5) - 2 (x^{2} + 25)}{(x + 5) (x - 5)} = 0$

Step 2.7.2.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} + 2 x \cdot - 5 - 10 (x + 5) - 2 (x^{2} + 25)}{(x + 5) (x - 5)} = 0$

Step 2.7.3

Multiply $- 5$ by $2$ .

$\frac{2 x^{2} - 10 x - 10 (x + 5) - 2 (x^{2} + 25)}{(x + 5) (x - 5)} = 0$

Step 2.7.4

Apply the distributive property.

$\frac{2 x^{2} - 10 x - 10 x - 10 \cdot 5 - 2 (x^{2} + 25)}{(x + 5) (x - 5)} = 0$

Step 2.7.5

Multiply $- 10$ by $5$ .

$\frac{2 x^{2} - 10 x - 10 x - 50 - 2 (x^{2} + 25)}{(x + 5) (x - 5)} = 0$

Step 2.7.6

Apply the distributive property.

$\frac{2 x^{2} - 10 x - 10 x - 50 - 2 x^{2} - 2 \cdot 25}{(x + 5) (x - 5)} = 0$

Step 2.7.7

Multiply $- 2$ by $25$ .

$\frac{2 x^{2} - 10 x - 10 x - 50 - 2 x^{2} - 50}{(x + 5) (x - 5)} = 0$

Step 2.8

Combine the opposite terms in $2 x^{2} - 10 x - 10 x - 50 - 2 x^{2} - 50$ .

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Step 2.8.1

Subtract $2 x^{2}$ from $2 x^{2}$ .

$\frac{- 10 x - 10 x - 50 + 0 - 50}{(x + 5) (x - 5)} = 0$

Step 2.8.2

Add $- 10 x - 10 x - 50$ and $0$ .

$\frac{- 10 x - 10 x - 50 - 50}{(x + 5) (x - 5)} = 0$

Step 2.9

Subtract $10 x$ from $- 10 x$ .

$\frac{- 20 x - 50 - 50}{(x + 5) (x - 5)} = 0$

Step 2.10

Subtract $50$ from $- 50$ .

$\frac{- 20 x - 100}{(x + 5) (x - 5)} = 0$

Step 2.11

Factor $20$ out of $- 20 x - 100$ .

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Step 2.11.1

Factor $20$ out of $- 20 x$ .

$\frac{20 (- x) - 100}{(x + 5) (x - 5)} = 0$

Step 2.11.2

Factor $20$ out of $- 100$ .

$\frac{20 (- x) + 20 (- 5)}{(x + 5) (x - 5)} = 0$

Step 2.11.3

Factor $20$ out of $20 (- x) + 20 (- 5)$ .

$\frac{20 (- x - 5)}{(x + 5) (x - 5)} = 0$

Step 2.12

Cancel the common factor of $- x - 5$ and $x + 5$ .

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Step 2.12.1

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

$\frac{20 (- (x) - 5)}{(x + 5) (x - 5)} = 0$

Step 2.12.2

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

$\frac{20 (- (x) - 1 (5))}{(x + 5) (x - 5)} = 0$

Step 2.12.3

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

$\frac{20 (- (x + 5))}{(x + 5) (x - 5)} = 0$

Step 2.12.4

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

$\frac{20 (- 1 (x + 5))}{(x + 5) (x - 5)} = 0$

Step 2.12.5

Cancel the common factor.

$\frac{20 (- 1 (x + 5))}{(x + 5) (x - 5)} = 0$

Step 2.12.6

Rewrite the expression.

$\frac{20 \cdot (- 1)}{x - 5} = 0$

Step 2.13

Multiply $20$ by $- 1$ .

$\frac{- 20}{x - 5} = 0$

Step 2.14

Move the negative in front of the fraction.

$- \frac{20}{x - 5} = 0$

Step 3

Set the numerator equal to zero.

$20 = 0$

Step 4

Since $20 \neq 0$ , there are no solutions.

No solution