Pre-Algebra Examples

\frac{8 a - 18}{3 a^{2} + 14 a + 8} + \frac{7}{3 a + 2}

$\frac{8 a - 18}{3 a^{2} + 14 a + 8} + \frac{7}{3 a + 2}$

Step 1

Simplify each term.

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Step 1.1

Factor

2

$2$ out of

8 a - 18

$8 a - 18$ .

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Step 1.1.1

Factor

2

$2$ out of

8 a

$8 a$ .

\frac{2 (4 a) - 18}{3 a^{2} + 14 a + 8} + \frac{7}{3 a + 2}

$\frac{2 (4 a) - 18}{3 a^{2} + 14 a + 8} + \frac{7}{3 a + 2}$

Step 1.1.2

Factor

2

$2$ out of

- 18

$- 18$ .

\frac{2 (4 a) + 2 (- 9)}{3 a^{2} + 14 a + 8} + \frac{7}{3 a + 2}

$\frac{2 (4 a) + 2 (- 9)}{3 a^{2} + 14 a + 8} + \frac{7}{3 a + 2}$

Step 1.1.3

Factor

2

$2$ out of

2 (4 a) + 2 (- 9)

$2 (4 a) + 2 (- 9)$ .

\frac{2 (4 a - 9)}{3 a^{2} + 14 a + 8} + \frac{7}{3 a + 2}

$\frac{2 (4 a - 9)}{3 a^{2} + 14 a + 8} + \frac{7}{3 a + 2}$

\frac{2 (4 a - 9)}{3 a^{2} + 14 a + 8} + \frac{7}{3 a + 2}

$\frac{2 (4 a - 9)}{3 a^{2} + 14 a + 8} + \frac{7}{3 a + 2}$

Step 1.2

Factor by grouping.

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Step 1.2.1

For a polynomial of the form

a x^{2} + b x + c

$a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is

a \cdot c = 3 \cdot 8 = 24

$a \cdot c = 3 \cdot 8 = 24$ and whose sum is

b = 14

$b = 14$ .

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Step 1.2.1.1

Factor

14

$14$ out of

14 a

$14 a$ .

\frac{2 (4 a - 9)}{3 a^{2} + 14 (a) + 8} + \frac{7}{3 a + 2}

$\frac{2 (4 a - 9)}{3 a^{2} + 14 (a) + 8} + \frac{7}{3 a + 2}$

Step 1.2.1.2

Rewrite

14

$14$ as

2

$2$ plus

12

$12$

\frac{2 (4 a - 9)}{3 a^{2} + (2 + 12) a + 8} + \frac{7}{3 a + 2}

$\frac{2 (4 a - 9)}{3 a^{2} + (2 + 12) a + 8} + \frac{7}{3 a + 2}$

Step 1.2.1.3

Apply the distributive property.

\frac{2 (4 a - 9)}{3 a^{2} + 2 a + 12 a + 8} + \frac{7}{3 a + 2}

$\frac{2 (4 a - 9)}{3 a^{2} + 2 a + 12 a + 8} + \frac{7}{3 a + 2}$

\frac{2 (4 a - 9)}{3 a^{2} + 2 a + 12 a + 8} + \frac{7}{3 a + 2}

$\frac{2 (4 a - 9)}{3 a^{2} + 2 a + 12 a + 8} + \frac{7}{3 a + 2}$

Step 1.2.2

Factor out the greatest common factor from each group.

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Step 1.2.2.1

Group the first two terms and the last two terms.

\frac{2 (4 a - 9)}{(3 a^{2} + 2 a) + 12 a + 8} + \frac{7}{3 a + 2}

$\frac{2 (4 a - 9)}{(3 a^{2} + 2 a) + 12 a + 8} + \frac{7}{3 a + 2}$

Step 1.2.2.2

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

\frac{2 (4 a - 9)}{a (3 a + 2) + 4 (3 a + 2)} + \frac{7}{3 a + 2}

$\frac{2 (4 a - 9)}{a (3 a + 2) + 4 (3 a + 2)} + \frac{7}{3 a + 2}$

Step 1.2.3

Factor the polynomial by factoring out the greatest common factor,

$3 a + 2$ .

$\frac{2 (4 a - 9)}{(3 a + 2) (a + 4)} + \frac{7}{3 a + 2}$

Step 2

To write

$\frac{7}{3 a + 2}$ as a fraction with a common denominator, multiply by

$\frac{a + 4}{a + 4}$ .

$\frac{2 (4 a - 9)}{(3 a + 2) (a + 4)} + \frac{7}{3 a + 2} \cdot \frac{a + 4}{a + 4}$

Step 3

Simplify terms.

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Step 3.1

Multiply

$\frac{7}{3 a + 2}$ by

$\frac{a + 4}{a + 4}$ .

$\frac{2 (4 a - 9)}{(3 a + 2) (a + 4)} + \frac{7 (a + 4)}{(3 a + 2) (a + 4)}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{2 (4 a - 9) + 7 (a + 4)}{(3 a + 2) (a + 4)}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{2 (4 a) + 2 \cdot - 9 + 7 (a + 4)}{(3 a + 2) (a + 4)}$

Step 4.2

Multiply

$4$ by

$2$ .

$\frac{8 a + 2 \cdot - 9 + 7 (a + 4)}{(3 a + 2) (a + 4)}$

Step 4.3

Multiply

$2$ by

$- 9$ .

$\frac{8 a - 18 + 7 (a + 4)}{(3 a + 2) (a + 4)}$

Step 4.4

Apply the distributive property.

$\frac{8 a - 18 + 7 a + 7 \cdot 4}{(3 a + 2) (a + 4)}$

Step 4.5

Multiply

$7$ by

$4$ .

$\frac{8 a - 18 + 7 a + 28}{(3 a + 2) (a + 4)}$

Step 4.6

Add

$8 a$ and

$7 a$ .

$\frac{15 a - 18 + 28}{(3 a + 2) (a + 4)}$

Step 4.7

Add

$- 18$ and

$28$ .

$\frac{15 a + 10}{(3 a + 2) (a + 4)}$

Step 4.8

Factor

$5$ out of

$15 a + 10$ .

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Step 4.8.1

Factor

$5$ out of

$15 a$ .

$\frac{5 (3 a) + 10}{(3 a + 2) (a + 4)}$

Step 4.8.2

Factor

$5$ out of

$10$ .

$\frac{5 (3 a) + 5 (2)}{(3 a + 2) (a + 4)}$

Step 4.8.3

Factor

$5$ out of

$5 (3 a) + 5 (2)$ .

$\frac{5 (3 a + 2)}{(3 a + 2) (a + 4)}$

Step 5

Cancel the common factor of

$3 a + 2$ .

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Step 5.1

Cancel the common factor.

$\frac{5 (3 a + 2)}{(3 a + 2) (a + 4)}$

Step 5.2

Rewrite the expression.

$\frac{5}{a + 4}$