Algebra Examples

$\frac{x^{2} + 10 x + 25}{x + 5} - \frac{x^{2} - 6}{x - 5}$

Step 1

Simplify each term.

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Step 1.1

Factor using the perfect square rule.

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Step 1.1.1

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

$\frac{x^{2} + 10 x + 5^{2}}{x + 5} - \frac{x^{2} - 6}{x - 5}$

Step 1.1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 x = 2 \cdot x \cdot 5$

Step 1.1.3

Rewrite the polynomial.

$\frac{x^{2} + 2 \cdot x \cdot 5 + 5^{2}}{x + 5} - \frac{x^{2} - 6}{x - 5}$

Step 1.1.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 5$ .

$\frac{{(x + 5)}^{2}}{x + 5} - \frac{x^{2} - 6}{x - 5}$

Step 1.2

Cancel the common factor of ${(x + 5)}^{2}$ and $x + 5$ .

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Step 1.2.1

Factor $x + 5$ out of ${(x + 5)}^{2}$ .

$\frac{(x + 5) (x + 5)}{x + 5} - \frac{x^{2} - 6}{x - 5}$

Step 1.2.2

Cancel the common factors.

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Step 1.2.2.1

Multiply by $1$ .

$\frac{(x + 5) (x + 5)}{(x + 5) \cdot 1} - \frac{x^{2} - 6}{x - 5}$

Step 1.2.2.2

Cancel the common factor.

$\frac{(x + 5) (x + 5)}{(x + 5) \cdot 1} - \frac{x^{2} - 6}{x - 5}$

Step 1.2.2.3

Rewrite the expression.

$\frac{x + 5}{1} - \frac{x^{2} - 6}{x - 5}$

Step 1.2.2.4

Divide $x + 5$ by $1$ .

$x + 5 - \frac{x^{2} - 6}{x - 5}$

Step 2

Find the common denominator.

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Step 2.1

Write $x$ as a fraction with denominator $1$ .

$\frac{x}{1} + 5 - \frac{x^{2} - 6}{x - 5}$

Step 2.2

Multiply $\frac{x}{1}$ by $\frac{x - 5}{x - 5}$ .

$\frac{x}{1} \cdot \frac{x - 5}{x - 5} + 5 - \frac{x^{2} - 6}{x - 5}$

Step 2.3

Multiply $\frac{x}{1}$ by $\frac{x - 5}{x - 5}$ .

$\frac{x (x - 5)}{x - 5} + 5 - \frac{x^{2} - 6}{x - 5}$

Step 2.4

Write $5$ as a fraction with denominator $1$ .

$\frac{x (x - 5)}{x - 5} + \frac{5}{1} - \frac{x^{2} - 6}{x - 5}$

Step 2.5

Multiply $\frac{5}{1}$ by $\frac{x - 5}{x - 5}$ .

$\frac{x (x - 5)}{x - 5} + \frac{5}{1} \cdot \frac{x - 5}{x - 5} - \frac{x^{2} - 6}{x - 5}$

Step 2.6

Multiply $\frac{5}{1}$ by $\frac{x - 5}{x - 5}$ .

$\frac{x (x - 5)}{x - 5} + \frac{5 (x - 5)}{x - 5} - \frac{x^{2} - 6}{x - 5}$

Step 3

Combine the numerators over the common denominator.

$\frac{x (x - 5) + 5 (x - 5) - (x^{2} - 6)}{x - 5}$

Step 4

Simplify each term.

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Step 4.1

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 5 + 5 (x - 5) - (x^{2} - 6)}{x - 5}$

Step 4.2

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot - 5 + 5 (x - 5) - (x^{2} - 6)}{x - 5}$

Step 4.3

Move $- 5$ to the left of $x$ .

$\frac{x^{2} - 5 \cdot x + 5 (x - 5) - (x^{2} - 6)}{x - 5}$

Step 4.4

Apply the distributive property.

$\frac{x^{2} - 5 x + 5 x + 5 \cdot - 5 - (x^{2} - 6)}{x - 5}$

Step 4.5

Multiply $5$ by $- 5$ .

$\frac{x^{2} - 5 x + 5 x - 25 - (x^{2} - 6)}{x - 5}$

Step 4.6

Apply the distributive property.

$\frac{x^{2} - 5 x + 5 x - 25 - x^{2} - - 6}{x - 5}$

Step 4.7

Multiply $- 1$ by $- 6$ .

$\frac{x^{2} - 5 x + 5 x - 25 - x^{2} + 6}{x - 5}$

Step 5

Simplify by adding terms.

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Step 5.1

Combine the opposite terms in $x^{2} - 5 x + 5 x - 25 - x^{2} + 6$ .

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Step 5.1.1

Add $- 5 x$ and $5 x$ .

$\frac{x^{2} + 0 - 25 - x^{2} + 6}{x - 5}$

Step 5.1.2

Add $x^{2}$ and $0$ .

$\frac{x^{2} - 25 - x^{2} + 6}{x - 5}$

Step 5.1.3

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

$\frac{0 - 25 + 6}{x - 5}$

Step 5.1.4

Subtract $25$ from $0$ .

$\frac{- 25 + 6}{x - 5}$

Step 5.2

Simplify the expression.

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Step 5.2.1

Add $- 25$ and $6$ .

$\frac{- 19}{x - 5}$

Step 5.2.2

Move the negative in front of the fraction.

$- \frac{19}{x - 5}$