Basic Math Examples

$\frac{9}{c^{2} - c - 6} + \frac{c}{c^{2} + 9 c + 14}$

Step 1

Simplify each term.

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Step 1.1

Factor $c^{2} - c - 6$ using the AC method.

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Step 1.1.1

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 1.1.2

Write the factored form using these integers.

$\frac{9}{(c - 3) (c + 2)} + \frac{c}{c^{2} + 9 c + 14}$

Step 1.2

Factor $c^{2} + 9 c + 14$ 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 $14$ and whose sum is $9$ .

$2, 7$

Step 1.2.2

Write the factored form using these integers.

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

Step 2

To write $\frac{9}{(c - 3) (c + 2)}$ as a fraction with a common denominator, multiply by $\frac{c + 7}{c + 7}$ .

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

Step 3

To write $\frac{c}{(c + 2) (c + 7)}$ as a fraction with a common denominator, multiply by $\frac{c - 3}{c - 3}$ .

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

Step 4

Write each expression with a common denominator of $(c - 3) (c + 2) (c + 7)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{9}{(c - 3) (c + 2)}$ by $\frac{c + 7}{c + 7}$ .

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

Step 4.2

Multiply $\frac{c}{(c + 2) (c + 7)}$ by $\frac{c - 3}{c - 3}$ .

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

Step 4.3

Reorder the factors of $(c - 3) (c + 2) (c + 7)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $9$ by $7$ .

$\frac{9 c + 63 + c (c - 3)}{(c + 2) (c + 7) (c - 3)}$

Step 6.3

Apply the distributive property.

$\frac{9 c + 63 + c \cdot c + c \cdot - 3}{(c + 2) (c + 7) (c - 3)}$

Step 6.4

Multiply $c$ by $c$ .

$\frac{9 c + 63 + c^{2} + c \cdot - 3}{(c + 2) (c + 7) (c - 3)}$

Step 6.5

Move $- 3$ to the left of $c$ .

$\frac{9 c + 63 + c^{2} - 3 \cdot c}{(c + 2) (c + 7) (c - 3)}$

Step 6.6

Subtract $3 c$ from $9 c$ .

$\frac{6 c + 63 + c^{2}}{(c + 2) (c + 7) (c - 3)}$

Step 6.7

Reorder terms.

$\frac{c^{2} + 6 c + 63}{(c + 2) (c + 7) (c - 3)}$