Pre-Algebra Examples

$\frac{9}{x^{2} - 12 x + 36} + \frac{x}{x^{2} - 36}$

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 $36$ as $6^{2}$ .

$\frac{9}{x^{2} - 12 x + 6^{2}} + \frac{x}{x^{2} - 36}$

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.

$12 x = 2 \cdot x \cdot 6$

Step 1.1.3

Rewrite the polynomial.

$\frac{9}{x^{2} - 2 \cdot x \cdot 6 + 6^{2}} + \frac{x}{x^{2} - 36}$

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 $b = 6$ .

$\frac{9}{{(x - 6)}^{2}} + \frac{x}{x^{2} - 36}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

Rewrite $36$ as $6^{2}$ .

$\frac{9}{{(x - 6)}^{2}} + \frac{x}{x^{2} - 6^{2}}$

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 $b = 6$ .

$\frac{9}{{(x - 6)}^{2}} + \frac{x}{(x + 6) (x - 6)}$

Step 2

To write $\frac{9}{{(x - 6)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{x + 6}{x + 6}$ .

$\frac{9}{{(x - 6)}^{2}} \cdot \frac{x + 6}{x + 6} + \frac{x}{(x + 6) (x - 6)}$

Step 3

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

$\frac{9}{{(x - 6)}^{2}} \cdot \frac{x + 6}{x + 6} + \frac{x}{(x + 6) (x - 6)} \cdot \frac{x - 6}{x - 6}$

Step 4

Write each expression with a common denominator of $(x + 6) {(x - 6)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{9}{{(x - 6)}^{2}}$ by $\frac{x + 6}{x + 6}$ .

$\frac{9 (x + 6)}{{(x - 6)}^{2} (x + 6)} + \frac{x}{(x + 6) (x - 6)} \cdot \frac{x - 6}{x - 6}$

Step 4.2

Multiply $\frac{x}{(x + 6) (x - 6)}$ by $\frac{x - 6}{x - 6}$ .

$\frac{9 (x + 6)}{{(x - 6)}^{2} (x + 6)} + \frac{x (x - 6)}{(x + 6) (x - 6) (x - 6)}$

Step 4.3

Raise $x - 6$ to the power of $1$ .

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

Step 4.4

Raise $x - 6$ to the power of $1$ .

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

Step 4.5

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

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

Step 4.6

Add $1$ and $1$ .

$\frac{9 (x + 6)}{{(x - 6)}^{2} (x + 6)} + \frac{x (x - 6)}{(x + 6) {(x - 6)}^{2}}$

Step 4.7

Reorder the factors of $(x + 6) {(x - 6)}^{2}$ .

$\frac{9 (x + 6)}{{(x - 6)}^{2} (x + 6)} + \frac{x (x - 6)}{{(x - 6)}^{2} (x + 6)}$

Step 5

Combine the numerators over the common denominator.

$\frac{9 (x + 6) + x (x - 6)}{{(x - 6)}^{2} (x + 6)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{9 x + 9 \cdot 6 + x (x - 6)}{{(x - 6)}^{2} (x + 6)}$

Step 6.2

Multiply $9$ by $6$ .

$\frac{9 x + 54 + x (x - 6)}{{(x - 6)}^{2} (x + 6)}$

Step 6.3

Apply the distributive property.

$\frac{9 x + 54 + x \cdot x + x \cdot - 6}{{(x - 6)}^{2} (x + 6)}$

Step 6.4

Multiply $x$ by $x$ .

$\frac{9 x + 54 + x^{2} + x \cdot - 6}{{(x - 6)}^{2} (x + 6)}$

Step 6.5

Move $- 6$ to the left of $x$ .

$\frac{9 x + 54 + x^{2} - 6 \cdot x}{{(x - 6)}^{2} (x + 6)}$

Step 6.6

Subtract $6 x$ from $9 x$ .

$\frac{3 x + 54 + x^{2}}{{(x - 6)}^{2} (x + 6)}$

Step 6.7

Reorder terms.

$\frac{x^{2} + 3 x + 54}{{(x - 6)}^{2} (x + 6)}$