Pre-Algebra Examples

$\frac{3}{x^{2} + 2 x - 15} - \frac{4}{x^{2} + 10 x + 25}$

Step 1

Simplify each term.

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Step 1.1

Factor $x^{2} + 2 x - 15$ 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 $- 15$ and whose sum is $2$ .

$- 3, 5$

Step 1.1.2

Write the factored form using these integers.

$\frac{3}{(x - 3) (x + 5)} - \frac{4}{x^{2} + 10 x + 25}$

Step 1.2

Factor using the perfect square rule.

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Step 1.2.1

Rewrite $25$ as $5^{2}$ .

$\frac{3}{(x - 3) (x + 5)} - \frac{4}{x^{2} + 10 x + 5^{2}}$

Step 1.2.2

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

$10 x = 2 \cdot x \cdot 5$

Step 1.2.3

Rewrite the polynomial.

$\frac{3}{(x - 3) (x + 5)} - \frac{4}{x^{2} + 2 \cdot x \cdot 5 + 5^{2}}$

Step 1.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 5$ .

$\frac{3}{(x - 3) (x + 5)} - \frac{4}{{(x + 5)}^{2}}$

Step 2

To write $\frac{3}{(x - 3) (x + 5)}$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

$\frac{3}{(x - 3) (x + 5)} \cdot \frac{x + 5}{x + 5} - \frac{4}{{(x + 5)}^{2}}$

Step 3

To write $- \frac{4}{{(x + 5)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{3}{(x - 3) (x + 5)} \cdot \frac{x + 5}{x + 5} - \frac{4}{{(x + 5)}^{2}} \cdot \frac{x - 3}{x - 3}$

Step 4

Write each expression with a common denominator of $(x - 3) {(x + 5)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{3}{(x - 3) (x + 5)}$ by $\frac{x + 5}{x + 5}$ .

$\frac{3 (x + 5)}{(x - 3) (x + 5) (x + 5)} - \frac{4}{{(x + 5)}^{2}} \cdot \frac{x - 3}{x - 3}$

Step 4.2

Raise $x + 5$ to the power of $1$ .

$\frac{3 (x + 5)}{(x - 3) ({(x + 5)}^{1} (x + 5))} - \frac{4}{{(x + 5)}^{2}} \cdot \frac{x - 3}{x - 3}$

Step 4.3

Raise $x + 5$ to the power of $1$ .

$\frac{3 (x + 5)}{(x - 3) ({(x + 5)}^{1} {(x + 5)}^{1})} - \frac{4}{{(x + 5)}^{2}} \cdot \frac{x - 3}{x - 3}$

Step 4.4

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

$\frac{3 (x + 5)}{(x - 3) {(x + 5)}^{1 + 1}} - \frac{4}{{(x + 5)}^{2}} \cdot \frac{x - 3}{x - 3}$

Step 4.5

Add $1$ and $1$ .

$\frac{3 (x + 5)}{(x - 3) {(x + 5)}^{2}} - \frac{4}{{(x + 5)}^{2}} \cdot \frac{x - 3}{x - 3}$

Step 4.6

Multiply $\frac{4}{{(x + 5)}^{2}}$ by $\frac{x - 3}{x - 3}$ .

$\frac{3 (x + 5)}{(x - 3) {(x + 5)}^{2}} - \frac{4 (x - 3)}{{(x + 5)}^{2} (x - 3)}$

Step 4.7

Reorder the factors of $(x - 3) {(x + 5)}^{2}$ .

$\frac{3 (x + 5)}{{(x + 5)}^{2} (x - 3)} - \frac{4 (x - 3)}{{(x + 5)}^{2} (x - 3)}$

Step 5

Combine the numerators over the common denominator.

$\frac{3 (x + 5) - 4 (x - 3)}{{(x + 5)}^{2} (x - 3)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{3 x + 3 \cdot 5 - 4 (x - 3)}{{(x + 5)}^{2} (x - 3)}$

Step 6.2

Multiply $3$ by $5$ .

$\frac{3 x + 15 - 4 (x - 3)}{{(x + 5)}^{2} (x - 3)}$

Step 6.3

Apply the distributive property.

$\frac{3 x + 15 - 4 x - 4 \cdot - 3}{{(x + 5)}^{2} (x - 3)}$

Step 6.4

Multiply $- 4$ by $- 3$ .

$\frac{3 x + 15 - 4 x + 12}{{(x + 5)}^{2} (x - 3)}$

Step 6.5

Subtract $4 x$ from $3 x$ .

$\frac{- x + 15 + 12}{{(x + 5)}^{2} (x - 3)}$

Step 6.6

Add $15$ and $12$ .

$\frac{- x + 27}{{(x + 5)}^{2} (x - 3)}$

Step 7

Simplify with factoring out.

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Step 7.1

Factor $- 1$ out of $- x$ .

$\frac{- (x) + 27}{{(x + 5)}^{2} (x - 3)}$

Step 7.2

Rewrite $27$ as $- 1 (- 27)$ .

$\frac{- (x) - 1 (- 27)}{{(x + 5)}^{2} (x - 3)}$

Step 7.3

Factor $- 1$ out of $- (x) - 1 (- 27)$ .

$\frac{- (x - 27)}{{(x + 5)}^{2} (x - 3)}$

Step 7.4

Simplify the expression.

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Step 7.4.1

Rewrite $- (x - 27)$ as $- 1 (x - 27)$ .

$\frac{- 1 (x - 27)}{{(x + 5)}^{2} (x - 3)}$

Step 7.4.2

Move the negative in front of the fraction.

$- \frac{x - 27}{{(x + 5)}^{2} (x - 3)}$