Pre-Algebra Examples

$\frac{x^{2} + 13 x - 20}{x^{2} - 25} + \frac{x - 1}{x + 5}$

Step 1

Simplify the denominator.

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Step 1.1

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

$\frac{x^{2} + 13 x - 20}{x^{2} - 5^{2}} + \frac{x - 1}{x + 5}$

Step 1.2

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

$\frac{x^{2} + 13 x - 20}{(x + 5) (x - 5)} + \frac{x - 1}{x + 5}$

Step 2

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

$\frac{x^{2} + 13 x - 20}{(x + 5) (x - 5)} + \frac{x - 1}{x + 5} \cdot \frac{x - 5}{x - 5}$

Step 3

Simplify terms.

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Step 3.1

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

$\frac{x^{2} + 13 x - 20}{(x + 5) (x - 5)} + \frac{(x - 1) (x - 5)}{(x + 5) (x - 5)}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{x^{2} + 13 x - 20 + (x - 1) (x - 5)}{(x + 5) (x - 5)}$

Step 4

Simplify the numerator.

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Step 4.1

Expand $(x - 1) (x - 5)$ using the FOIL Method.

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Step 4.1.1

Apply the distributive property.

$\frac{x^{2} + 13 x - 20 + x (x - 5) - 1 (x - 5)}{(x + 5) (x - 5)}$

Step 4.1.2

Apply the distributive property.

$\frac{x^{2} + 13 x - 20 + x \cdot x + x \cdot - 5 - 1 (x - 5)}{(x + 5) (x - 5)}$

Step 4.1.3

Apply the distributive property.

$\frac{x^{2} + 13 x - 20 + x \cdot x + x \cdot - 5 - 1 x - 1 \cdot - 5}{(x + 5) (x - 5)}$

Step 4.2

Simplify and combine like terms.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $x$ by $x$ .

$\frac{x^{2} + 13 x - 20 + x^{2} + x \cdot - 5 - 1 x - 1 \cdot - 5}{(x + 5) (x - 5)}$

Step 4.2.1.2

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

$\frac{x^{2} + 13 x - 20 + x^{2} - 5 \cdot x - 1 x - 1 \cdot - 5}{(x + 5) (x - 5)}$

Step 4.2.1.3

Rewrite $- 1 x$ as $- x$ .

$\frac{x^{2} + 13 x - 20 + x^{2} - 5 x - x - 1 \cdot - 5}{(x + 5) (x - 5)}$

Step 4.2.1.4

Multiply $- 1$ by $- 5$ .

$\frac{x^{2} + 13 x - 20 + x^{2} - 5 x - x + 5}{(x + 5) (x - 5)}$

Step 4.2.2

Subtract $x$ from $- 5 x$ .

$\frac{x^{2} + 13 x - 20 + x^{2} - 6 x + 5}{(x + 5) (x - 5)}$

Step 4.3

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

$\frac{2 x^{2} + 13 x - 20 - 6 x + 5}{(x + 5) (x - 5)}$

Step 4.4

Subtract $6 x$ from $13 x$ .

$\frac{2 x^{2} + 7 x - 20 + 5}{(x + 5) (x - 5)}$

Step 4.5

Add $- 20$ and $5$ .

$\frac{2 x^{2} + 7 x - 15}{(x + 5) (x - 5)}$

Step 4.6

Factor by grouping.

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Step 4.6.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 15 = - 30$ and whose sum is $b = 7$ .

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Step 4.6.1.1

Factor $7$ out of $7 x$ .

$\frac{2 x^{2} + 7 (x) - 15}{(x + 5) (x - 5)}$

Step 4.6.1.2

Rewrite $7$ as $- 3$ plus $10$

$\frac{2 x^{2} + (- 3 + 10) x - 15}{(x + 5) (x - 5)}$

Step 4.6.1.3

Apply the distributive property.

$\frac{2 x^{2} - 3 x + 10 x - 15}{(x + 5) (x - 5)}$

Step 4.6.2

Factor out the greatest common factor from each group.

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Step 4.6.2.1

Group the first two terms and the last two terms.

$\frac{(2 x^{2} - 3 x) + 10 x - 15}{(x + 5) (x - 5)}$

Step 4.6.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 4.6.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 3$ .

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

Step 5

Cancel the common factor of $x + 5$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

$\frac{2 x - 3}{x - 5}$