Pre-Algebra Examples

$\frac{x^{2} - 5 x}{x^{2} + 3 x - 4} + \frac{5 x - 16}{x^{2} + 3 x - 4}$

Step 1

Simplify terms.

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Step 1.1

Combine the numerators over the common denominator.

$\frac{x^{2} - 5 x + 5 x - 16}{x^{2} + 3 x - 4}$

Step 1.2

Combine the opposite terms in $x^{2} - 5 x + 5 x - 16$ .

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Step 1.2.1

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

$\frac{x^{2} + 0 - 16}{x^{2} + 3 x - 4}$

Step 1.2.2

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

$\frac{x^{2} - 16}{x^{2} + 3 x - 4}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $16$ as $4^{2}$ .

$\frac{x^{2} - 4^{2}}{x^{2} + 3 x - 4}$

Step 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 $b = 4$ .

$\frac{(x + 4) (x - 4)}{x^{2} + 3 x - 4}$

Step 3

Factor $x^{2} + 3 x - 4$ using the AC method.

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Step 3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 4$ and whose sum is $3$ .

$- 1, 4$

Step 3.2

Write the factored form using these integers.

$\frac{(x + 4) (x - 4)}{(x - 1) (x + 4)}$

Step 4

Cancel the common factor of $x + 4$ .

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Step 4.1

Cancel the common factor.

$\frac{(x + 4) (x - 4)}{(x - 1) (x + 4)}$

Step 4.2

Rewrite the expression.

$\frac{x - 4}{x - 1}$