Find the Excluded Values ((8x+32)/(x^2+8x+16))÷((x^2-6x)/(x^2-2x-24))

$\frac{8 x + 32}{x^{2} + 8 x + 16} \div \frac{x^{2} - 6 x}{x^{2} - 2 x - 24}$

Set the denominator in $\frac{8 x + 32}{x^{2} + 8 x + 16}$ equal to $0$ to find where the expression is undefined.

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

Solve for $x$ .

Factor using the perfect square rule.

Rewrite $16$ as $4^{2}$ .

$x^{2} + 8 x + 4^{2} = 0$

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

$8 x = 2 \cdot x \cdot 4$

Rewrite the polynomial.

$x^{2} + 2 \cdot x \cdot 4 + 4^{2} = 0$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 4$ .

${(x + 4)}^{2} = 0$

Set the $x + 4$ equal to $0$ .

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

Set the denominator in $\frac{x^{2} - 6 x}{x^{2} - 2 x - 24}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 2 x - 24 = 0$

Solve for $x$ .

Factor $x^{2} - 2 x - 24$ 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 $- 24$ and whose sum is $- 2$ .

$- 6, 4$

Write the factored form using these integers.

$(x - 6) (x + 4) = 0$

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

$x - 6 = 0$

$x + 4 = 0$

Set $x - 6$ equal to $0$ and solve for $x$ .

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

Set $x + 4$ equal to $0$ and solve for $x$ .

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

The final solution is all the values that make $(x - 6) (x + 4) = 0$ true.

$x = 6, - 4$

Set the denominator in $\frac{\frac{8 x + 32}{x^{2} + 8 x + 16}}{\frac{x^{2} - 6 x}{x^{2} - 2 x - 24}}$ equal to $0$ to find where the expression is undefined.

$\frac{x^{2} - 6 x}{x^{2} - 2 x - 24} = 0$

Solve for $x$ .

Set the numerator equal to zero.

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

Solve the equation for $x$ .

Factor $x$ out of $x^{2} - 6 x$ .

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

$x \cdot x - 6 x = 0$

Factor $x$ out of $- 6 x$ .

$x \cdot x + x \cdot - 6 = 0$

Factor $x$ out of $x \cdot x + x \cdot - 6$ .

$x (x - 6) = 0$

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

$x = 0$

$x - 6 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x - 6$ equal to $0$ and solve for $x$ .

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

The final solution is all the values that make $x (x - 6) = 0$ true.

$x = 0, 6$

Exclude the solutions that do not make $\frac{x^{2} - 6 x}{x^{2} - 2 x - 24} = 0$ true.

$x = 0$

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 $0$ .

$x = - 4, x = 0, x = 6$

Find the Excluded Values ((8x+32)/(x^2+8x+16))÷((x^2-6x)/(x^2-2x-24))

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