Pre-Algebra Examples

$\frac{36 - g^{2}}{30 - 5 g} \div \frac{g^{2} + g - 30}{7 g + 6}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{36 - g^{2}}{30 - 5 g} \cdot \frac{7 g + 6}{g^{2} + g - 30}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $36$ as $6^{2}$ .

$\frac{6^{2} - g^{2}}{30 - 5 g} \cdot \frac{7 g + 6}{g^{2} + g - 30}$

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 $a = 6$ and $b = g$ .

$\frac{(6 + g) (6 - g)}{30 - 5 g} \cdot \frac{7 g + 6}{g^{2} + g - 30}$

Step 3

Simplify terms.

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Step 3.1

Factor $5$ out of $30 - 5 g$ .

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Step 3.1.1

Factor $5$ out of $30$ .

$\frac{(6 + g) (6 - g)}{5 (6) - 5 g} \cdot \frac{7 g + 6}{g^{2} + g - 30}$

Step 3.1.2

Factor $5$ out of $- 5 g$ .

$\frac{(6 + g) (6 - g)}{5 (6) + 5 (- g)} \cdot \frac{7 g + 6}{g^{2} + g - 30}$

Step 3.1.3

Factor $5$ out of $5 (6) + 5 (- g)$ .

$\frac{(6 + g) (6 - g)}{5 (6 - g)} \cdot \frac{7 g + 6}{g^{2} + g - 30}$

Step 3.2

Cancel the common factor of $6 - g$ .

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Step 3.2.1

Cancel the common factor.

$\frac{(6 + g) (6 - g)}{5 (6 - g)} \cdot \frac{7 g + 6}{g^{2} + g - 30}$

Step 3.2.2

Rewrite the expression.

$\frac{6 + g}{5} \cdot \frac{7 g + 6}{g^{2} + g - 30}$

Step 4

Factor $g^{2} + g - 30$ using the AC method.

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Step 4.1

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 4.2

Write the factored form using these integers.

$\frac{6 + g}{5} \cdot \frac{7 g + 6}{(g - 5) (g + 6)}$

Step 5

Simplify terms.

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Step 5.1

Multiply $\frac{6 + g}{5}$ by $\frac{7 g + 6}{(g - 5) (g + 6)}$ .

$\frac{(6 + g) (7 g + 6)}{5 ((g - 5) (g + 6))}$

Step 5.2

Cancel the common factor of $6 + g$ and $g + 6$ .

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Step 5.2.1

Reorder terms.

$\frac{(g + 6) (7 g + 6)}{5 ((g - 5) (g + 6))}$

Step 5.2.2

Cancel the common factor.

$\frac{(g + 6) (7 g + 6)}{5 ((g - 5) (g + 6))}$

Step 5.2.3

Rewrite the expression.

$\frac{7 g + 6}{5 (g - 5)}$