Algebra Examples

$\frac{3}{x + 6} + \frac{x}{x^{2} + x - 30}$

Step 1

Factor $x^{2} + x - 30$ 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 $- 30$ and whose sum is $1$ .

$- 5, 6$

Step 1.2

Write the factored form using these integers.

$\frac{3}{x + 6} + \frac{x}{(x - 5) (x + 6)}$

Step 2

To write $\frac{3}{x + 6}$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

$\frac{3}{x + 6} \cdot \frac{x - 5}{x - 5} + \frac{x}{(x - 5) (x + 6)}$

Step 3

Write each expression with a common denominator of $(x + 6) (x - 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{3}{x + 6}$ by $\frac{x - 5}{x - 5}$ .

$\frac{3 (x - 5)}{(x + 6) (x - 5)} + \frac{x}{(x - 5) (x + 6)}$

Step 3.2

Reorder the factors of $(x - 5) (x + 6)$ .

$\frac{3 (x - 5)}{(x + 6) (x - 5)} + \frac{x}{(x + 6) (x - 5)}$

Step 4

Combine the numerators over the common denominator.

$\frac{3 (x - 5) + x}{(x + 6) (x - 5)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{3 x + 3 \cdot - 5 + x}{(x + 6) (x - 5)}$

Step 5.2

Multiply $3$ by $- 5$ .

$\frac{3 x - 15 + x}{(x + 6) (x - 5)}$

Step 5.3

Add $3 x$ and $x$ .

$\frac{4 x - 15}{(x + 6) (x - 5)}$