Find the Excluded Values ((x+8)/(x^2-6x-7))/((x^2+16x+64)/(x+1))

$\frac{\frac{x + 8}{x^{2} - 6 x - 7}}{\frac{x^{2} + 16 x + 64}{x + 1}}$

Step 1

Set the denominator in $\frac{x + 8}{x^{2} - 6 x - 7}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 6 x - 7 = 0$

Step 2

Solve for $x$ .

Factor $x^{2} - 6 x - 7$ 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 $- 7$ and whose sum is $- 6$ .

$- 7, 1$

Write the factored form using these integers.

$(x - 7) (x + 1) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 7 = 0$

$x + 1 = 0$

Set $x - 7$ equal to $0$ and solve for $x$ .

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

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

Set $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

The final solution is all the values that make $(x - 7) (x + 1) = 0$ true.

$x = 7, - 1$

Step 3

Set the denominator in $\frac{x^{2} + 16 x + 64}{x + 1}$ equal to $0$ to find where the expression is undefined.

$x + 1 = 0$

Step 4

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 5

Set the denominator in $\frac{\frac{x + 8}{x^{2} - 6 x - 7}}{\frac{x^{2} + 16 x + 64}{x + 1}}$ equal to $0$ to find where the expression is undefined.

$\frac{x^{2} + 16 x + 64}{x + 1} = 0$

Step 6

Solve for $x$ .

Set the numerator equal to zero.

$x^{2} + 16 x + 64 = 0$

Solve the equation for $x$ .

Factor using the perfect square rule.

Rewrite $64$ as $8^{2}$ .

$x^{2} + 16 x + 8^{2} = 0$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$16 x = 2 \cdot x \cdot 8$

Rewrite the polynomial.

$x^{2} + 2 \cdot x \cdot 8 + 8^{2} = 0$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 8$ .

${(x + 8)}^{2} = 0$

Set the $x + 8$ equal to $0$ .

$x + 8 = 0$

Subtract $8$ from both sides of the equation.

$x = - 8$

Step 7

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = - 8, x = - 1, x = 7$

Step 8

Find the Excluded Values ((x+8)/(x^2-6x-7))/((x^2+16x+64)/(x+1))

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