Basic Math Examples

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

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

Simplify the numerator.

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Step 2.1

Raise $6$ to the power of $2$ .

$\frac{36 + 7 a - 3}{2 a^{2} - a - 6} \cdot \frac{a - 2}{a + 5}$

Step 2.2

Subtract $3$ from $36$ .

$\frac{7 a + 33}{2 a^{2} - a - 6} \cdot \frac{a - 2}{a + 5}$

Step 3

Factor by grouping.

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Step 3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 6 = - 12$ and whose sum is $b = - 1$ .

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Step 3.1.1

Factor $- 1$ out of $- a$ .

$\frac{7 a + 33}{2 a^{2} - (a) - 6} \cdot \frac{a - 2}{a + 5}$

Step 3.1.2

Rewrite $- 1$ as $3$ plus $- 4$

$\frac{7 a + 33}{2 a^{2} + (3 - 4) a - 6} \cdot \frac{a - 2}{a + 5}$

Step 3.1.3

Apply the distributive property.

$\frac{7 a + 33}{2 a^{2} + 3 a - 4 a - 6} \cdot \frac{a - 2}{a + 5}$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$\frac{7 a + 33}{(2 a^{2} + 3 a) - 4 a - 6} \cdot \frac{a - 2}{a + 5}$

Step 3.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{7 a + 33}{a (2 a + 3) - 2 (2 a + 3)} \cdot \frac{a - 2}{a + 5}$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $2 a + 3$ .

$\frac{7 a + 33}{(2 a + 3) (a - 2)} \cdot \frac{a - 2}{a + 5}$

Step 4

Cancel the common factor of $a - 2$ .

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Step 4.1

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

$\frac{7 a + 33}{(a - 2) (2 a + 3)} \cdot \frac{a - 2}{a + 5}$

Step 4.2

Cancel the common factor.

$\frac{7 a + 33}{(a - 2) (2 a + 3)} \cdot \frac{a - 2}{a + 5}$

Step 4.3

Rewrite the expression.

$\frac{7 a + 33}{2 a + 3} \cdot \frac{1}{a + 5}$

Step 5

Multiply $\frac{7 a + 33}{2 a + 3}$ by $\frac{1}{a + 5}$ .

$\frac{7 a + 33}{(2 a + 3) (a + 5)}$