Algebra Examples

$\frac{w + 4}{w^{2} + w - 30} + \frac{8}{w - 5}$

Step 1

Factor $w^{2} + w - 30$ 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.

$\frac{w + 4}{(w - 5) (w + 6)} + \frac{8}{w - 5}$

Step 2

To write $\frac{8}{w - 5}$ as a fraction with a common denominator, multiply by $\frac{w + 6}{w + 6}$ .

$\frac{w + 4}{(w - 5) (w + 6)} + \frac{8}{w - 5} \cdot \frac{w + 6}{w + 6}$

Step 3

Simplify terms.

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Step 3.1

Multiply $\frac{8}{w - 5}$ by $\frac{w + 6}{w + 6}$ .

$\frac{w + 4}{(w - 5) (w + 6)} + \frac{8 (w + 6)}{(w - 5) (w + 6)}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{w + 4 + 8 (w + 6)}{(w - 5) (w + 6)}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{w + 4 + 8 w + 8 \cdot 6}{(w - 5) (w + 6)}$

Step 4.2

Multiply $8$ by $6$ .

$\frac{w + 4 + 8 w + 48}{(w - 5) (w + 6)}$

Step 4.3

Add $w$ and $8 w$ .

$\frac{9 w + 4 + 48}{(w - 5) (w + 6)}$

Step 4.4

Add $4$ and $48$ .

$\frac{9 w + 52}{(w - 5) (w + 6)}$