Pre-Algebra Examples

\frac{x}{x - 3} + \frac{21}{x^{2} - 4} = \frac{18}{x^{2} - 9}

$\frac{x}{x - 3} + \frac{21}{x^{2} - 4} = \frac{18}{x^{2} - 9}$

Step 1

Simplify

$\frac{x}{x - 3} + \frac{21}{x^{2} - 4}$ .

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Step 1.1

Simplify the denominator.

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Step 1.1.1

Rewrite

$4$ as

$2^{2}$ .

$\frac{x}{x - 3} + \frac{21}{x^{2} - 2^{2}} = \frac{18}{x^{2} - 9}$

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

$b = 2$ .

$\frac{x}{x - 3} + \frac{21}{(x + 2) (x - 2)} = \frac{18}{x^{2} - 9}$

Step 1.2

To write

$\frac{x}{x - 3}$ as a fraction with a common denominator, multiply by

$\frac{(x + 2) (x - 2)}{(x + 2) (x - 2)}$ .

$\frac{x}{x - 3} \cdot \frac{(x + 2) (x - 2)}{(x + 2) (x - 2)} + \frac{21}{(x + 2) (x - 2)} = \frac{18}{x^{2} - 9}$

Step 1.3

To write

$\frac{21}{(x + 2) (x - 2)}$ as a fraction with a common denominator, multiply by

$\frac{x - 3}{x - 3}$ .

$\frac{x}{x - 3} \cdot \frac{(x + 2) (x - 2)}{(x + 2) (x - 2)} + \frac{21}{(x + 2) (x - 2)} \cdot \frac{x - 3}{x - 3} = \frac{18}{x^{2} - 9}$

Step 1.4

Write each expression with a common denominator of

$(x - 3) (x + 2) (x - 2)$ , by multiplying each by an appropriate factor of

$1$ .

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Step 1.4.1

Multiply

$\frac{x}{x - 3}$ by

$\frac{(x + 2) (x - 2)}{(x + 2) (x - 2)}$ .

$\frac{x ((x + 2) (x - 2))}{(x - 3) ((x + 2) (x - 2))} + \frac{21}{(x + 2) (x - 2)} \cdot \frac{x - 3}{x - 3} = \frac{18}{x^{2} - 9}$

Step 1.4.2

Multiply

$\frac{21}{(x + 2) (x - 2)}$ by

$\frac{x - 3}{x - 3}$ .

$\frac{x ((x + 2) (x - 2))}{(x - 3) ((x + 2) (x - 2))} + \frac{21 (x - 3)}{(x + 2) (x - 2) (x - 3)} = \frac{18}{x^{2} - 9}$

Step 1.4.3

Reorder the factors of

$(x - 3) ((x + 2) (x - 2))$ .

$\frac{x ((x + 2) (x - 2))}{(x + 2) (x - 2) (x - 3)} + \frac{21 (x - 3)}{(x + 2) (x - 2) (x - 3)} = \frac{18}{x^{2} - 9}$

Step 1.5

Combine the numerators over the common denominator.

$\frac{x ((x + 2) (x - 2)) + 21 (x - 3)}{(x + 2) (x - 2) (x - 3)} = \frac{18}{x^{2} - 9}$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Expand

$(x + 2) (x - 2)$ using the FOIL Method.

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Step 1.6.1.1

Apply the distributive property.

$\frac{x (x (x - 2) + 2 (x - 2)) + 21 (x - 3)}{(x + 2) (x - 2) (x - 3)} = \frac{18}{x^{2} - 9}$

Step 1.6.1.2

Apply the distributive property.

$\frac{x (x \cdot x + x \cdot - 2 + 2 (x - 2)) + 21 (x - 3)}{(x + 2) (x - 2) (x - 3)} = \frac{18}{x^{2} - 9}$

Step 1.6.1.3

Apply the distributive property.

$\frac{x (x \cdot x + x \cdot - 2 + 2 x + 2 \cdot - 2) + 21 (x - 3)}{(x + 2) (x - 2) (x - 3)} = \frac{18}{x^{2} - 9}$

Step 1.6.2

Combine the opposite terms in

$x \cdot x + x \cdot - 2 + 2 x + 2 \cdot - 2$ .

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Step 1.6.2.1

Reorder the factors in the terms

$x \cdot - 2$ and

$2 x$ .

$\frac{x (x \cdot x - 2 x + 2 x + 2 \cdot - 2) + 21 (x - 3)}{(x + 2) (x - 2) (x - 3)} = \frac{18}{x^{2} - 9}$

Step 1.6.2.2

Add

$- 2 x$ and

$2 x$ .

$\frac{x (x \cdot x + 0 + 2 \cdot - 2) + 21 (x - 3)}{(x + 2) (x - 2) (x - 3)} = \frac{18}{x^{2} - 9}$

Step 1.6.2.3

Add

$x \cdot x$ and

$0$ .

$\frac{x (x \cdot x + 2 \cdot - 2) + 21 (x - 3)}{(x + 2) (x - 2) (x - 3)} = \frac{18}{x^{2} - 9}$

Step 1.6.3

Simplify each term.

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Step 1.6.3.1

Multiply

$x$ by

$x$ .

$\frac{x (x^{2} + 2 \cdot - 2) + 21 (x - 3)}{(x + 2) (x - 2) (x - 3)} = \frac{18}{x^{2} - 9}$

Step 1.6.3.2

Multiply

$2$ by

$- 2$ .

$\frac{x (x^{2} - 4) + 21 (x - 3)}{(x + 2) (x - 2) (x - 3)} = \frac{18}{x^{2} - 9}$

Step 1.6.4

Apply the distributive property.

$\frac{x \cdot x^{2} + x \cdot - 4 + 21 (x - 3)}{(x + 2) (x - 2) (x - 3)} = \frac{18}{x^{2} - 9}$

Step 1.6.5

Multiply

$x$ by

$x^{2}$ by adding the exponents.

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Step 1.6.5.1

Multiply

$x$ by

$x^{2}$ .

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Step 1.6.5.1.1

Raise

$x$ to the power of

$1$ .

$\frac{x^{1} x^{2} + x \cdot - 4 + 21 (x - 3)}{(x + 2) (x - 2) (x - 3)} = \frac{18}{x^{2} - 9}$

Step 1.6.5.1.2

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{1 + 2} + x \cdot - 4 + 21 (x - 3)}{(x + 2) (x - 2) (x - 3)} = \frac{18}{x^{2} - 9}$

Step 1.6.5.2

Add

$1$ and

$2$ .

$\frac{x^{3} + x \cdot - 4 + 21 (x - 3)}{(x + 2) (x - 2) (x - 3)} = \frac{18}{x^{2} - 9}$

Step 1.6.6

Move

$- 4$ to the left of

$x$ .

$\frac{x^{3} - 4 \cdot x + 21 (x - 3)}{(x + 2) (x - 2) (x - 3)} = \frac{18}{x^{2} - 9}$

Step 1.6.7

Apply the distributive property.

$\frac{x^{3} - 4 x + 21 x + 21 \cdot - 3}{(x + 2) (x - 2) (x - 3)} = \frac{18}{x^{2} - 9}$

Step 1.6.8

Multiply

$21$ by

$- 3$ .

$\frac{x^{3} - 4 x + 21 x - 63}{(x + 2) (x - 2) (x - 3)} = \frac{18}{x^{2} - 9}$

Step 1.6.9

Add

$- 4 x$ and

$21 x$ .

$\frac{x^{3} + 17 x - 63}{(x + 2) (x - 2) (x - 3)} = \frac{18}{x^{2} - 9}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite

$9$ as

$3^{2}$ .

$\frac{x^{3} + 17 x - 63}{(x + 2) (x - 2) (x - 3)} = \frac{18}{x^{2} - 3^{2}}$

Step 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

$b = 3$ .

$\frac{x^{3} + 17 x - 63}{(x + 2) (x - 2) (x - 3)} = \frac{18}{(x + 3) (x - 3)}$

Step 3

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx - 4.16642404$

Step 4