Find the Holes in the Graph f(x)=(5x^2-45x+100)/(-4x^2+20x)

$f (x) = \frac{5 x^{2} - 45 x + 100}{- 4 x^{2} + 20 x}$

Factor $5 x^{2} - 45 x + 100$ .

Factor $5$ out of $5 x^{2} - 45 x + 100$ .

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

$f (x) = \frac{5 (x^{2}) - 45 x + 100}{- 4 x^{2} + 20 x}$

Factor $5$ out of $- 45 x$ .

$f (x) = \frac{5 (x^{2}) + 5 (- 9 x) + 100}{- 4 x^{2} + 20 x}$

Factor $5$ out of $100$ .

$f (x) = \frac{5 x^{2} + 5 (- 9 x) + 5 \cdot 20}{- 4 x^{2} + 20 x}$

Factor $5$ out of $5 x^{2} + 5 (- 9 x)$ .

$f (x) = \frac{5 (x^{2} - 9 x) + 5 \cdot 20}{- 4 x^{2} + 20 x}$

Factor $5$ out of $5 (x^{2} - 9 x) + 5 \cdot 20$ .

$f (x) = \frac{5 (x^{2} - 9 x + 20)}{- 4 x^{2} + 20 x}$

Factor.

Factor $x^{2} - 9 x + 20$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $20$ and whose sum is $- 9$ .

$- 5, - 4$

Write the factored form using these integers.

$f (x) = \frac{5 ((x - 5) (x - 4))}{- 4 x^{2} + 20 x}$

Remove unnecessary parentheses.

$f (x) = \frac{5 (x - 5) (x - 4)}{- 4 x^{2} + 20 x}$

Factor $4 x$ out of $- 4 x^{2} + 20 x$ .

Factor $4 x$ out of $- 4 x^{2}$ .

$f (x) = \frac{5 (x - 5) (x - 4)}{4 x (- x) + 20 x}$

Factor $4 x$ out of $20 x$ .

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

Factor $4 x$ out of $4 x (- x) + 4 x (5)$ .

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

Cancel the common factor of $x - 5$ and $- x + 5$ .

Factor $- 1$ out of $x$ .

$f (x) = \frac{5 (- 1 (- x) - 5) (x - 4)}{4 x (- x + 5)}$

Rewrite $- 5$ as $- 1 (5)$ .

$f (x) = \frac{5 (- 1 (- x) - 1 \cdot 5) (x - 4)}{4 x (- x + 5)}$

Factor $- 1$ out of $- 1 (- x) - 1 (5)$ .

$f (x) = \frac{5 (- 1 (- x + 5)) (x - 4)}{4 x (- x + 5)}$

Cancel the common factor.

$f (x) = \frac{5 (- 1 (- x + 5)) (x - 4)}{4 x (- x + 5)}$

Rewrite the expression.

$f (x) = \frac{5 \cdot ((- 1) (x - 4))}{4 x}$

Combine exponents.

Factor out negative.

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

Multiply $5$ by $- 1$ .

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

Move the negative in front of the fraction.

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

Multiply $5$ by $x - 4$ .

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

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

$- x + 5$

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

Set $- x + 5$ equal to $0$ .

$- x + 5 = 0$

Solve $- x + 5 = 0$ for $x$ .

Subtract $5$ from both sides of the equation.

$- x = - 5$

Divide each term in $- x = - 5$ by $- 1$ and simplify.

Divide each term in $- x = - 5$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 5}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 5}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 5}{- 1}$

Simplify the right side.

Divide $- 5$ by $- 1$ .

$x = 5$

Substitute $5$ for $x$ in $- \frac{5 (x - 4)}{4 x}$ and simplify.

Substitute $5$ for $x$ to find the $y$ coordinate of the hole.

$- \frac{5 (5 - 4)}{4 \cdot 5}$

Simplify.

Cancel the common factor of $5$ .

Cancel the common factor.

$- \frac{5 (5 - 4)}{4 \cdot 5}$

Rewrite the expression.

$- \frac{5 - 4}{4}$

Subtract $4$ from $5$ .

$- \frac{1}{4}$

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

$(5, - \frac{1}{4})$

Find the Holes in the Graph f(x)=(5x^2-45x+100)/(-4x^2+20x)

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