Find the Holes in the Graph f(x)=(5x^2-10x)/(4x-8)

$f (x) = \frac{5 x^{2} - 10 x}{4 x - 8}$

Step 1

Factor $5 x$ out of $5 x^{2} - 10 x$ .

Factor $5 x$ out of $5 x^{2}$ .

$f (x) = \frac{5 x (x) - 10 x}{4 x - 8}$

Factor $5 x$ out of $- 10 x$ .

$f (x) = \frac{5 x (x) + 5 x (- 2)}{4 x - 8}$

Factor $5 x$ out of $5 x (x) + 5 x (- 2)$ .

$f (x) = \frac{5 x (x - 2)}{4 x - 8}$

Step 2

Factor $4$ out of $4 x - 8$ .

Factor $4$ out of $4 x$ .

$f (x) = \frac{5 x (x - 2)}{4 (x) - 8}$

Factor $4$ out of $- 8$ .

$f (x) = \frac{5 x (x - 2)}{4 x + 4 \cdot - 2}$

Factor $4$ out of $4 x + 4 \cdot - 2$ .

$f (x) = \frac{5 x (x - 2)}{4 (x - 2)}$

Step 3

Cancel the common factor of $x - 2$ .

Cancel the common factor.

$f (x) = \frac{5 x (x - 2)}{4 (x - 2)}$

Rewrite the expression.

$f (x) = \frac{5 x}{4}$

Step 4

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

$x - 2$

Step 5

To find the coordinates of the holes, set each factor that was cancelled equal to $0$ , solve, and substitute back in to $\frac{5 x}{4}$ .

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Substitute $2$ for $x$ in $\frac{5 x}{4}$ and simplify.

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

$\frac{5 \cdot 2}{4}$

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $5 \cdot 2$ .

$\frac{2 \cdot 5}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$\frac{2 \cdot 5}{2 \cdot 2}$

Cancel the common factor.

$\frac{2 \cdot 5}{2 \cdot 2}$

Rewrite the expression.

$\frac{5}{2}$

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

$(2, \frac{5}{2})$

Step 6

Find the Holes in the Graph f(x)=(5x^2-10x)/(4x-8)

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