Algebra Examples

$\frac{a^{2} - 5 a - 14}{a - 7} \div \frac{a^{2} - a - 6}{5 a^{3} + 30 a^{2}}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{a^{2} - 5 a - 14}{a - 7} \cdot \frac{5 a^{3} + 30 a^{2}}{a^{2} - a - 6}$

Step 2

Factor $a^{2} - 5 a - 14$ using the AC method.

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Step 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 $- 14$ and whose sum is $- 5$ .

$- 7, 2$

Step 2.2

Write the factored form using these integers.

$\frac{(a - 7) (a + 2)}{a - 7} \cdot \frac{5 a^{3} + 30 a^{2}}{a^{2} - a - 6}$

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $a - 7$ .

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Step 3.1.1

Cancel the common factor.

$\frac{(a - 7) (a + 2)}{a - 7} \cdot \frac{5 a^{3} + 30 a^{2}}{a^{2} - a - 6}$

Step 3.1.2

Divide $a + 2$ by $1$ .

$(a + 2) \frac{5 a^{3} + 30 a^{2}}{a^{2} - a - 6}$

Step 3.2

Factor $5 a^{2}$ out of $5 a^{3} + 30 a^{2}$ .

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Step 3.2.1

Factor $5 a^{2}$ out of $5 a^{3}$ .

$(a + 2) \frac{5 a^{2} (a) + 30 a^{2}}{a^{2} - a - 6}$

Step 3.2.2

Factor $5 a^{2}$ out of $30 a^{2}$ .

$(a + 2) \frac{5 a^{2} (a) + 5 a^{2} (6)}{a^{2} - a - 6}$

Step 3.2.3

Factor $5 a^{2}$ out of $5 a^{2} (a) + 5 a^{2} (6)$ .

$(a + 2) \frac{5 a^{2} (a + 6)}{a^{2} - a - 6}$

Step 4

Factor $a^{2} - a - 6$ 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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 4.2

Write the factored form using these integers.

$(a + 2) \frac{5 a^{2} (a + 6)}{(a - 3) (a + 2)}$

Step 5

Cancel the common factor of $a + 2$ .

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Step 5.1

Factor $a + 2$ out of $(a - 3) (a + 2)$ .

$(a + 2) \frac{5 a^{2} (a + 6)}{(a + 2) (a - 3)}$

Step 5.2

Cancel the common factor.

$(a + 2) \frac{5 a^{2} (a + 6)}{(a + 2) (a - 3)}$

Step 5.3

Rewrite the expression.

$\frac{5 a^{2} (a + 6)}{a - 3}$