Find the Holes in the Graph y=(x+2)/(x^2-4)

$y = \frac{x + 2}{x^{2} - 4}$

Step 1

Factor $x^{2} - 4$ .

Rewrite $4$ as $2^{2}$ .

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

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 $b = 2$ .

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

Step 2

Cancel the common factor of $x + 2$ .

Cancel the common factor.

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

Rewrite the expression.

$y = \frac{1}{x - 2}$

Step 3

To find the holes in the graph, look at the denominator factors that were cancelled.

$x + 2$

Step 4

To find the coordinates of the holes, set each factor that was cancelled equal to $0$ , solve, and substitute back in to $\frac{1}{x - 2}$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Substitute $- 2$ for $x$ in $\frac{1}{x - 2}$ and simplify.

Substitute $- 2$ for $x$ to find the $y$ coordinate of the hole.

$\frac{1}{- 2 - 2}$

Simplify.

Subtract $2$ from $- 2$ .

$\frac{1}{- 4}$

Move the negative in front of the fraction.

$- \frac{1}{4}$

The holes in the graph are the points where any of the cancelled factors are equal to $0$ .

$(- 2, - \frac{1}{4})$

Step 5