Precalculus Examples

$f (x) = \frac{X^{2} - 1}{x + 2}$

Step 1

Subtract $\frac{X^{2} - 1}{x + 2}$ from both sides of the equation.

$y - \frac{X^{2} - 1}{x + 2} = 0$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $1$ as $1^{2}$ .

$y - \frac{X^{2} - 1^{2}}{x + 2} = 0$

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 $b = 1$ .

$y - \frac{(X + 1) (X - 1)}{x + 2} = 0$

Step 3

To write $y$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$\frac{y (x + 2)}{x + 2} - \frac{(X + 1) (X - 1)}{x + 2} = 0$

Step 4

Combine the numerators over the common denominator.

$\frac{y (x + 2) - (X + 1) (X - 1)}{x + 2} = 0$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{y x + y \cdot 2 - (X + 1) (X - 1)}{x + 2} = 0$

Step 5.2

Move $2$ to the left of $y$ .

$\frac{y x + 2 \cdot y - (X + 1) (X - 1)}{x + 2} = 0$

Step 5.3

Apply the distributive property.

$\frac{y x + 2 y + (- X - 1 \cdot 1) (X - 1)}{x + 2} = 0$

Step 5.4

Multiply $- 1$ by $1$ .

$\frac{y x + 2 y + (- X - 1) (X - 1)}{x + 2} = 0$

Step 5.5

Expand $(- X - 1) (X - 1)$ using the FOIL Method.

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Step 5.5.1

Apply the distributive property.

$\frac{y x + 2 y - X (X - 1) - 1 (X - 1)}{x + 2} = 0$

Step 5.5.2

Apply the distributive property.

$\frac{y x + 2 y - X \cdot X - X \cdot - 1 - 1 (X - 1)}{x + 2} = 0$

Step 5.5.3

Apply the distributive property.

$\frac{y x + 2 y - X \cdot X - X \cdot - 1 - 1 X - 1 \cdot - 1}{x + 2} = 0$

Step 5.6

Simplify and combine like terms.

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Step 5.6.1

Simplify each term.

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Step 5.6.1.1

Multiply $X$ by $X$ by adding the exponents.

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Step 5.6.1.1.1

Move $X$ .

$\frac{y x + 2 y - (X \cdot X) - X \cdot - 1 - 1 X - 1 \cdot - 1}{x + 2} = 0$

Step 5.6.1.1.2

Multiply $X$ by $X$ .

$\frac{y x + 2 y - X^{2} - X \cdot - 1 - 1 X - 1 \cdot - 1}{x + 2} = 0$

Step 5.6.1.2

Multiply $- X \cdot - 1$ .

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Step 5.6.1.2.1

Multiply $- 1$ by $- 1$ .

$\frac{y x + 2 y - X^{2} + 1 X - 1 X - 1 \cdot - 1}{x + 2} = 0$

Step 5.6.1.2.2

Multiply $X$ by $1$ .

$\frac{y x + 2 y - X^{2} + X - 1 X - 1 \cdot - 1}{x + 2} = 0$

Step 5.6.1.3

Rewrite $- 1 X$ as $- X$ .

$\frac{y x + 2 y - X^{2} + X - X - 1 \cdot - 1}{x + 2} = 0$

Step 5.6.1.4

Multiply $- 1$ by $- 1$ .

$\frac{y x + 2 y - X^{2} + X - X + 1}{x + 2} = 0$

Step 5.6.2

Subtract $X$ from $X$ .

$\frac{y x + 2 y - X^{2} + 0 + 1}{x + 2} = 0$

Step 5.6.3

Add $- X^{2}$ and $0$ .

$\frac{y x + 2 y - X^{2} + 1}{x + 2} = 0$