Find the Holes in the Graph H(x)=(4x^2-80x+364)/(x^2-18x+77)

$H (x) = \frac{4 x^{2} - 80 x + 364}{x^{2} - 18 x + 77}$

Step 1

Factor $4 x^{2} - 80 x + 364$ .

Factor $4$ out of $4 x^{2} - 80 x + 364$ .

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

$H (x) = \frac{4 (x^{2}) - 80 x + 364}{x^{2} - 18 x + 77}$

Factor $4$ out of $- 80 x$ .

$H (x) = \frac{4 (x^{2}) + 4 (- 20 x) + 364}{x^{2} - 18 x + 77}$

Factor $4$ out of $364$ .

$H (x) = \frac{4 x^{2} + 4 (- 20 x) + 4 \cdot 91}{x^{2} - 18 x + 77}$

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

$H (x) = \frac{4 (x^{2} - 20 x) + 4 \cdot 91}{x^{2} - 18 x + 77}$

Factor $4$ out of $4 (x^{2} - 20 x) + 4 \cdot 91$ .

$H (x) = \frac{4 (x^{2} - 20 x + 91)}{x^{2} - 18 x + 77}$

Factor.

Factor $x^{2} - 20 x + 91$ 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 $91$ and whose sum is $- 20$ .

$- 13, - 7$

Write the factored form using these integers.

$H (x) = \frac{4 ((x - 13) (x - 7))}{x^{2} - 18 x + 77}$

Remove unnecessary parentheses.

$H (x) = \frac{4 (x - 13) (x - 7)}{x^{2} - 18 x + 77}$

Step 2

Factor $x^{2} - 18 x + 77$ 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 $77$ and whose sum is $- 18$ .

$- 11, - 7$

Write the factored form using these integers.

$H (x) = \frac{4 (x - 13) (x - 7)}{(x - 11) (x - 7)}$

Step 3

Cancel the common factor of $x - 7$ .

Cancel the common factor.

$H (x) = \frac{4 (x - 13) (x - 7)}{(x - 11) (x - 7)}$

Rewrite the expression.

$H (x) = \frac{4 (x - 13)}{x - 11}$

Step 4

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

$x - 7$

Step 5

To find the coordinates of the holes, set each factor that was cancelled equal to $0$ , solve, and substitute back in to $\frac{4 (x - 13)}{x - 11}$ .

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

Substitute $7$ for $x$ in $\frac{4 (x - 13)}{x - 11}$ and simplify.

Substitute $7$ for $x$ to find the $y$ coordinate of the hole.

$\frac{4 (7 - 13)}{7 - 11}$

Simplify.

Subtract $13$ from $7$ .

$\frac{4 \cdot - 6}{7 - 11}$

Subtract $11$ from $7$ .

$\frac{4 \cdot - 6}{- 4}$

Multiply $4$ by $- 6$ .

$\frac{- 24}{- 4}$

Divide $- 24$ by $- 4$ .

$6$

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

$(7, 6)$

Step 6

Find the Holes in the Graph H(x)=(4x^2-80x+364)/(x^2-18x+77)

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