Find the Holes in the Graph (2x^2-18)/(x^2-3x)

$\frac{2 x^{2} - 18}{x^{2} - 3 x}$

Step 1

Factor $2 x^{2} - 18$ .

Factor $2$ out of $2 x^{2} - 18$ .

Factor $2$ out of $2 x^{2}$ .

$\frac{2 (x^{2}) - 18}{x^{2} - 3 x}$

Factor $2$ out of $- 18$ .

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

Factor $2$ out of $2 x^{2} + 2 \cdot - 9$ .

$\frac{2 (x^{2} - 9)}{x^{2} - 3 x}$

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

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

Factor.

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 $b = 3$ .

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

Remove unnecessary parentheses.

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

Step 2

Factor $x$ out of $x^{2} - 3 x$ .

Factor $x$ out of $x^{2}$ .

$\frac{2 (x + 3) (x - 3)}{x \cdot x - 3 x}$

Factor $x$ out of $- 3 x$ .

$\frac{2 (x + 3) (x - 3)}{x \cdot x + x \cdot - 3}$

Factor $x$ out of $x \cdot x + x \cdot - 3$ .

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

Step 3

Cancel the common factor of $x - 3$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{2 (x + 3)}{x}$

Step 4

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

$x - 3$

Step 5

To find the coordinates of the holes, set each factor that was cancelled equal to $0$ , solve, and substitute back in to $\frac{2 (x + 3)}{x}$ .

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

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Substitute $3$ for $x$ in $\frac{2 (x + 3)}{x}$ and simplify.

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

$\frac{2 (3 + 3)}{3}$

Simplify.

Cancel the common factor of $3 + 3$ and $3$ .

Factor $3$ out of $2 (3 + 3)$ .

$\frac{3 (2 (1 + 1))}{3}$

Cancel the common factors.

Factor $3$ out of $3$ .

$\frac{3 (2 (1 + 1))}{3 (1)}$

Cancel the common factor.

$\frac{3 (2 (1 + 1))}{3 \cdot 1}$

Rewrite the expression.

$\frac{2 (1 + 1)}{1}$

Divide $2 (1 + 1)$ by $1$ .

$2 (1 + 1)$

Add $1$ and $1$ .

$2 \cdot 2$

Multiply $2$ by $2$ .

$4$

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

$(3, 4)$

Step 6

Find the Holes in the Graph (2x^2-18)/(x^2-3x)

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