Precalculus Examples

$\frac{6 x}{x^{2} + 6 x + 9} + \frac{7}{x^{2} + 5 x + 6}$

Step 1

Simplify each term.

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Step 1.1

Factor using the perfect square rule.

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Step 1.1.1

Rewrite $9$ as $3^{2}$ .

$\frac{6 x}{x^{2} + 6 x + 3^{2}} + \frac{7}{x^{2} + 5 x + 6}$

Step 1.1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 x = 2 \cdot x \cdot 3$

Step 1.1.3

Rewrite the polynomial.

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

Step 1.1.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 3$ .

$\frac{6 x}{{(x + 3)}^{2}} + \frac{7}{x^{2} + 5 x + 6}$

Step 1.2

Factor $x^{2} + 5 x + 6$ 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 $6$ and whose sum is $5$ .

$2, 3$

Step 1.2.2

Write the factored form using these integers.

$\frac{6 x}{{(x + 3)}^{2}} + \frac{7}{(x + 2) (x + 3)}$

Step 2

To write $\frac{6 x}{{(x + 3)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$\frac{6 x}{{(x + 3)}^{2}} \cdot \frac{x + 2}{x + 2} + \frac{7}{(x + 2) (x + 3)}$

Step 3

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

$\frac{6 x}{{(x + 3)}^{2}} \cdot \frac{x + 2}{x + 2} + \frac{7}{(x + 2) (x + 3)} \cdot \frac{x + 3}{x + 3}$

Step 4

Write each expression with a common denominator of $(x + 2) {(x + 3)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{6 x}{{(x + 3)}^{2}}$ by $\frac{x + 2}{x + 2}$ .

$\frac{6 x (x + 2)}{{(x + 3)}^{2} (x + 2)} + \frac{7}{(x + 2) (x + 3)} \cdot \frac{x + 3}{x + 3}$

Step 4.2

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

$\frac{6 x (x + 2)}{{(x + 3)}^{2} (x + 2)} + \frac{7 (x + 3)}{(x + 2) (x + 3) (x + 3)}$

Step 4.3

Raise $x + 3$ to the power of $1$ .

$\frac{6 x (x + 2)}{{(x + 3)}^{2} (x + 2)} + \frac{7 (x + 3)}{(x + 2) ({(x + 3)}^{1} (x + 3))}$

Step 4.4

Raise $x + 3$ to the power of $1$ .

$\frac{6 x (x + 2)}{{(x + 3)}^{2} (x + 2)} + \frac{7 (x + 3)}{(x + 2) ({(x + 3)}^{1} {(x + 3)}^{1})}$

Step 4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{6 x (x + 2)}{{(x + 3)}^{2} (x + 2)} + \frac{7 (x + 3)}{(x + 2) {(x + 3)}^{1 + 1}}$

Step 4.6

Add $1$ and $1$ .

$\frac{6 x (x + 2)}{{(x + 3)}^{2} (x + 2)} + \frac{7 (x + 3)}{(x + 2) {(x + 3)}^{2}}$

Step 4.7

Reorder the factors of $(x + 2) {(x + 3)}^{2}$ .

$\frac{6 x (x + 2)}{{(x + 3)}^{2} (x + 2)} + \frac{7 (x + 3)}{{(x + 3)}^{2} (x + 2)}$

Step 5

Combine the numerators over the common denominator.

$\frac{6 x (x + 2) + 7 (x + 3)}{{(x + 3)}^{2} (x + 2)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{6 x \cdot x + 6 x \cdot 2 + 7 (x + 3)}{{(x + 3)}^{2} (x + 2)}$

Step 6.2

Multiply $x$ by $x$ by adding the exponents.

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Step 6.2.1

Move $x$ .

$\frac{6 (x \cdot x) + 6 x \cdot 2 + 7 (x + 3)}{{(x + 3)}^{2} (x + 2)}$

Step 6.2.2

Multiply $x$ by $x$ .

$\frac{6 x^{2} + 6 x \cdot 2 + 7 (x + 3)}{{(x + 3)}^{2} (x + 2)}$

Step 6.3

Multiply $2$ by $6$ .

$\frac{6 x^{2} + 12 x + 7 (x + 3)}{{(x + 3)}^{2} (x + 2)}$

Step 6.4

Apply the distributive property.

$\frac{6 x^{2} + 12 x + 7 x + 7 \cdot 3}{{(x + 3)}^{2} (x + 2)}$

Step 6.5

Multiply $7$ by $3$ .

$\frac{6 x^{2} + 12 x + 7 x + 21}{{(x + 3)}^{2} (x + 2)}$

Step 6.6

Add $12 x$ and $7 x$ .

$\frac{6 x^{2} + 19 x + 21}{{(x + 3)}^{2} (x + 2)}$