Algebra Examples

$\frac{x + 4}{x^{2} + 6 x + 9} - \frac{1}{x^{2} - 9}$

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 $3^{2}$ .

$\frac{x + 4}{x^{2} + 6 x + 3^{2}} - \frac{1}{x^{2} - 9}$

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.

$\frac{x + 4}{x^{2} + 2 \cdot x \cdot 3 + 3^{2}} - \frac{1}{x^{2} - 9}$

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 $b = 3$ .

$\frac{x + 4}{{(x + 3)}^{2}} - \frac{1}{x^{2} - 9}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

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

$\frac{x + 4}{{(x + 3)}^{2}} - \frac{1}{x^{2} - 3^{2}}$

Step 1.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

$\frac{x + 4}{{(x + 3)}^{2}} - \frac{1}{(x + 3) (x - 3)}$

Step 2

To write $\frac{x + 4}{{(x + 3)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{x + 4}{{(x + 3)}^{2}} \cdot \frac{x - 3}{x - 3} - \frac{1}{(x + 3) (x - 3)}$

Step 3

To write $- \frac{1}{(x + 3) (x - 3)}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

$\frac{x + 4}{{(x + 3)}^{2}} \cdot \frac{x - 3}{x - 3} - \frac{1}{(x + 3) (x - 3)} \cdot \frac{x + 3}{x + 3}$

Step 4

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

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Step 4.1

Multiply $\frac{x + 4}{{(x + 3)}^{2}}$ by $\frac{x - 3}{x - 3}$ .

$\frac{(x + 4) (x - 3)}{{(x + 3)}^{2} (x - 3)} - \frac{1}{(x + 3) (x - 3)} \cdot \frac{x + 3}{x + 3}$

Step 4.2

Multiply $\frac{1}{(x + 3) (x - 3)}$ by $\frac{x + 3}{x + 3}$ .

$\frac{(x + 4) (x - 3)}{{(x + 3)}^{2} (x - 3)} - \frac{x + 3}{(x + 3) (x - 3) (x + 3)}$

Step 4.3

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

$\frac{(x + 4) (x - 3)}{{(x + 3)}^{2} (x - 3)} - \frac{x + 3}{{(x + 3)}^{1} (x + 3) (x - 3)}$

Step 4.4

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

$\frac{(x + 4) (x - 3)}{{(x + 3)}^{2} (x - 3)} - \frac{x + 3}{{(x + 3)}^{1} {(x + 3)}^{1} (x - 3)}$

Step 4.5

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

$\frac{(x + 4) (x - 3)}{{(x + 3)}^{2} (x - 3)} - \frac{x + 3}{{(x + 3)}^{1 + 1} (x - 3)}$

Step 4.6

Add $1$ and $1$ .

$\frac{(x + 4) (x - 3)}{{(x + 3)}^{2} (x - 3)} - \frac{x + 3}{{(x + 3)}^{2} (x - 3)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(x + 4) (x - 3) - (x + 3)}{{(x + 3)}^{2} (x - 3)}$

Step 6

Simplify the numerator.

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Step 6.1

Expand $(x + 4) (x - 3)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

$\frac{x (x - 3) + 4 (x - 3) - (x + 3)}{{(x + 3)}^{2} (x - 3)}$

Step 6.1.2

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 3 + 4 (x - 3) - (x + 3)}{{(x + 3)}^{2} (x - 3)}$

Step 6.1.3

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 3 + 4 x + 4 \cdot - 3 - (x + 3)}{{(x + 3)}^{2} (x - 3)}$

Step 6.2

Simplify and combine like terms.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot - 3 + 4 x + 4 \cdot - 3 - (x + 3)}{{(x + 3)}^{2} (x - 3)}$

Step 6.2.1.2

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

$\frac{x^{2} - 3 \cdot x + 4 x + 4 \cdot - 3 - (x + 3)}{{(x + 3)}^{2} (x - 3)}$

Step 6.2.1.3

Multiply $4$ by $- 3$ .

$\frac{x^{2} - 3 x + 4 x - 12 - (x + 3)}{{(x + 3)}^{2} (x - 3)}$

Step 6.2.2

Add $- 3 x$ and $4 x$ .

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

Step 6.3

Apply the distributive property.

$\frac{x^{2} + x - 12 - x - 1 \cdot 3}{{(x + 3)}^{2} (x - 3)}$

Step 6.4

Multiply $- 1$ by $3$ .

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

Step 6.5

Subtract $x$ from $x$ .

$\frac{x^{2} + 0 - 12 - 3}{{(x + 3)}^{2} (x - 3)}$

Step 6.6

Add $x^{2}$ and $0$ .

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

Step 6.7

Subtract $3$ from $- 12$ .

$\frac{x^{2} - 15}{{(x + 3)}^{2} (x - 3)}$