Algebra Examples

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

Step 1

Factor $x^{2} - 9$ .

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Step 1.1

Rewrite $9$ as $3^{2}$ .

$y = \frac{x + 3}{x^{2} - 3^{2}}$

Step 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 $b = 3$ .

$y = \frac{x + 3}{(x + 3) (x - 3)}$

Step 2

Cancel the common factor of $x + 3$ .

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Step 2.1

Cancel the common factor.

$y = \frac{x + 3}{(x + 3) (x - 3)}$

Step 2.2

Rewrite the expression.

$y = \frac{1}{x - 3}$

Step 3

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

$x + 3$

Step 4

To find the coordinates of the holes, set each factor that was cancelled equal to $0$ , solve, and substitute back in to $\frac{1}{x - 3}$ .

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Step 4.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 4.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 4.3

Substitute $- 3$ for $x$ in $\frac{1}{x - 3}$ and simplify.

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Step 4.3.1

Substitute $- 3$ for $x$ to find the $y$ coordinate of the hole.

$\frac{1}{- 3 - 3}$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Subtract $3$ from $- 3$ .

$\frac{1}{- 6}$

Step 4.3.2.2

Move the negative in front of the fraction.

$- \frac{1}{6}$

Step 4.4

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

$(- 3, - \frac{1}{6})$

Step 5