Algebra Examples

$\frac{x}{x - 3} - \frac{3}{x + 4} + \frac{7}{x^{2} + x - 12}$

Step 1

Factor $x^{2} + x - 12$ using the AC method.

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Step 1.1

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 $- 12$ and whose sum is $1$ .

$- 3, 4$

Step 1.2

Write the factored form using these integers.

$\frac{x}{x - 3} - \frac{3}{x + 4} + \frac{7}{(x - 3) (x + 4)}$

Step 2

To write $\frac{x}{x - 3}$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

$\frac{x}{x - 3} \cdot \frac{x + 4}{x + 4} - \frac{3}{x + 4} + \frac{7}{(x - 3) (x + 4)}$

Step 3

To write $- \frac{3}{x + 4}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{x}{x - 3} \cdot \frac{x + 4}{x + 4} - \frac{3}{x + 4} \cdot \frac{x - 3}{x - 3} + \frac{7}{(x - 3) (x + 4)}$

Step 4

Write each expression with a common denominator of $(x - 3) (x + 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{x}{x - 3}$ by $\frac{x + 4}{x + 4}$ .

$\frac{x (x + 4)}{(x - 3) (x + 4)} - \frac{3}{x + 4} \cdot \frac{x - 3}{x - 3} + \frac{7}{(x - 3) (x + 4)}$

Step 4.2

Multiply $\frac{3}{x + 4}$ by $\frac{x - 3}{x - 3}$ .

$\frac{x (x + 4)}{(x - 3) (x + 4)} - \frac{3 (x - 3)}{(x + 4) (x - 3)} + \frac{7}{(x - 3) (x + 4)}$

Step 4.3

Reorder the factors of $(x - 3) (x + 4)$ .

$\frac{x (x + 4)}{(x + 4) (x - 3)} - \frac{3 (x - 3)}{(x + 4) (x - 3)} + \frac{7}{(x - 3) (x + 4)}$

Step 5

Simplify terms.

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Step 5.1

Combine the numerators over the common denominator.

$\frac{x (x + 4) - 3 (x - 3)}{(x + 4) (x - 3)} + \frac{7}{(x - 3) (x + 4)}$

Step 5.2

Combine the numerators over the common denominator.

$\frac{x (x + 4) - 3 (x - 3) + 7}{(x + 4) (x - 3)}$

Step 6

Simplify each term.

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Step 6.1

Apply the distributive property.

$\frac{x \cdot x + x \cdot 4 - 3 (x - 3) + 7}{(x + 4) (x - 3)}$

Step 6.2

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot 4 - 3 (x - 3) + 7}{(x + 4) (x - 3)}$

Step 6.3

Move $4$ to the left of $x$ .

$\frac{x^{2} + 4 \cdot x - 3 (x - 3) + 7}{(x + 4) (x - 3)}$

Step 6.4

Apply the distributive property.

$\frac{x^{2} + 4 x - 3 x - 3 \cdot - 3 + 7}{(x + 4) (x - 3)}$

Step 6.5

Multiply $- 3$ by $- 3$ .

$\frac{x^{2} + 4 x - 3 x + 9 + 7}{(x + 4) (x - 3)}$

Step 7

Simplify by adding terms.

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Step 7.1

Subtract $3 x$ from $4 x$ .

$\frac{x^{2} + x + 9 + 7}{(x + 4) (x - 3)}$

Step 7.2

Add $9$ and $7$ .

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