Basic Math Examples

$\frac{2 r - 2}{r^{2} + 4 r - 32} + \frac{3}{r + 8} \cdot \frac{3}{r - 4}$

Step 1

Simplify each term.

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Step 1.1

Factor $2$ out of $2 r - 2$ .

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Step 1.1.1

Factor $2$ out of $2 r$ .

$\frac{2 (r) - 2}{r^{2} + 4 r - 32} + \frac{3}{r + 8} \cdot \frac{3}{r - 4}$

Step 1.1.2

Factor $2$ out of $- 2$ .

$\frac{2 (r) + 2 (- 1)}{r^{2} + 4 r - 32} + \frac{3}{r + 8} \cdot \frac{3}{r - 4}$

Step 1.1.3

Factor $2$ out of $2 (r) + 2 (- 1)$ .

$\frac{2 (r - 1)}{r^{2} + 4 r - 32} + \frac{3}{r + 8} \cdot \frac{3}{r - 4}$

Step 1.2

Factor $r^{2} + 4 r - 32$ using the AC method.

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Step 1.2.1

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 $- 32$ and whose sum is $4$ .

$- 4, 8$

Step 1.2.2

Write the factored form using these integers.

$\frac{2 (r - 1)}{(r - 4) (r + 8)} + \frac{3}{r + 8} \cdot \frac{3}{r - 4}$

Step 1.3

Multiply $\frac{3}{r + 8} \cdot \frac{3}{r - 4}$ .

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Step 1.3.1

Multiply $\frac{3}{r + 8}$ by $\frac{3}{r - 4}$ .

$\frac{2 (r - 1)}{(r - 4) (r + 8)} + \frac{3 \cdot 3}{(r + 8) (r - 4)}$

Step 1.3.2

Multiply $3$ by $3$ .

$\frac{2 (r - 1)}{(r - 4) (r + 8)} + \frac{9}{(r + 8) (r - 4)}$

Step 2

Combine the numerators over the common denominator.

$\frac{2 (r - 1) + 9}{(r + 8) (r - 4)}$

Step 3

Simplify each term.

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Step 3.1

Apply the distributive property.

$\frac{2 r + 2 \cdot - 1 + 9}{(r + 8) (r - 4)}$

Step 3.2

Multiply $2$ by $- 1$ .

$\frac{2 r - 2 + 9}{(r + 8) (r - 4)}$

Step 4

Add $- 2$ and $9$ .

$\frac{2 r + 7}{(r + 8) (r - 4)}$