Pre-Algebra Examples

$\frac{8 x}{x^{2} - 49} + \frac{5}{x^{2} - 4 x - 21}$

Step 1

Simplify each term.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

Rewrite $49$ as $7^{2}$ .

$\frac{8 x}{x^{2} - 7^{2}} + \frac{5}{x^{2} - 4 x - 21}$

Step 1.1.2

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 $b = 7$ .

$\frac{8 x}{(x + 7) (x - 7)} + \frac{5}{x^{2} - 4 x - 21}$

Step 1.2

Factor $x^{2} - 4 x - 21$ 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 $- 21$ and whose sum is $- 4$ .

$- 7, 3$

Step 1.2.2

Write the factored form using these integers.

$\frac{8 x}{(x + 7) (x - 7)} + \frac{5}{(x - 7) (x + 3)}$

Step 2

To write $\frac{8 x}{(x + 7) (x - 7)}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

$\frac{8 x}{(x + 7) (x - 7)} \cdot \frac{x + 3}{x + 3} + \frac{5}{(x - 7) (x + 3)}$

Step 3

To write $\frac{5}{(x - 7) (x + 3)}$ as a fraction with a common denominator, multiply by $\frac{x + 7}{x + 7}$ .

$\frac{8 x}{(x + 7) (x - 7)} \cdot \frac{x + 3}{x + 3} + \frac{5}{(x - 7) (x + 3)} \cdot \frac{x + 7}{x + 7}$

Step 4

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

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Step 4.1

Multiply $\frac{8 x}{(x + 7) (x - 7)}$ by $\frac{x + 3}{x + 3}$ .

$\frac{8 x (x + 3)}{(x + 7) (x - 7) (x + 3)} + \frac{5}{(x - 7) (x + 3)} \cdot \frac{x + 7}{x + 7}$

Step 4.2

Multiply $\frac{5}{(x - 7) (x + 3)}$ by $\frac{x + 7}{x + 7}$ .

$\frac{8 x (x + 3)}{(x + 7) (x - 7) (x + 3)} + \frac{5 (x + 7)}{(x - 7) (x + 3) (x + 7)}$

Step 4.3

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

$\frac{8 x (x + 3)}{(x + 3) (x + 7) (x - 7)} + \frac{5 (x + 7)}{(x - 7) (x + 3) (x + 7)}$

Step 4.4

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

$\frac{8 x (x + 3)}{(x + 3) (x + 7) (x - 7)} + \frac{5 (x + 7)}{(x + 3) (x + 7) (x - 7)}$

Step 5

Combine the numerators over the common denominator.

$\frac{8 x (x + 3) + 5 (x + 7)}{(x + 3) (x + 7) (x - 7)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{8 x \cdot x + 8 x \cdot 3 + 5 (x + 7)}{(x + 3) (x + 7) (x - 7)}$

Step 6.2

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

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Step 6.2.1

Move $x$ .

$\frac{8 (x \cdot x) + 8 x \cdot 3 + 5 (x + 7)}{(x + 3) (x + 7) (x - 7)}$

Step 6.2.2

Multiply $x$ by $x$ .

$\frac{8 x^{2} + 8 x \cdot 3 + 5 (x + 7)}{(x + 3) (x + 7) (x - 7)}$

Step 6.3

Multiply $3$ by $8$ .

$\frac{8 x^{2} + 24 x + 5 (x + 7)}{(x + 3) (x + 7) (x - 7)}$

Step 6.4

Apply the distributive property.

$\frac{8 x^{2} + 24 x + 5 x + 5 \cdot 7}{(x + 3) (x + 7) (x - 7)}$

Step 6.5

Multiply $5$ by $7$ .

$\frac{8 x^{2} + 24 x + 5 x + 35}{(x + 3) (x + 7) (x - 7)}$

Step 6.6

Add $24 x$ and $5 x$ .

$\frac{8 x^{2} + 29 x + 35}{(x + 3) (x + 7) (x - 7)}$