Find the Excluded Values ((x^2-9x+18)/(x^2+9x+18))÷((x^2-36)/(x^2-9))

$\frac{x^{2} - 9 x + 18}{x^{2} + 9 x + 18} \div \frac{x^{2} - 36}{x^{2} - 9}$

Step 1

Set the denominator in $\frac{x^{2} - 9 x + 18}{x^{2} + 9 x + 18}$ equal to $0$ to find where the expression is undefined.

$x^{2} + 9 x + 18 = 0$

Step 2

Solve for $x$ .

Factor $x^{2} + 9 x + 18$ 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 $18$ and whose sum is $9$ .

$3, 6$

Write the factored form using these integers.

$(x + 3) (x + 6) = 0$

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

$x + 3 = 0$

$x + 6 = 0$

Set $x + 3$ equal to $0$ and solve for $x$ .

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

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

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

$x + 6 = 0$

Subtract $6$ from both sides of the equation.

$x = - 6$

The final solution is all the values that make $(x + 3) (x + 6) = 0$ true.

$x = - 3, - 6$

Step 3

Set the denominator in $\frac{x^{2} - 36}{x^{2} - 9}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 9 = 0$

Step 4

Solve for $x$ .

Add $9$ to both sides of the equation.

$x^{2} = 9$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{9}$

Simplify $\pm \sqrt{9}$ .

Rewrite $9$ as $3^{2}$ .

$x = \pm \sqrt{3^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 3$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = 3$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 3$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 3, - 3$

Step 5

Set the denominator in $\frac{\frac{x^{2} - 9 x + 18}{x^{2} + 9 x + 18}}{\frac{x^{2} - 36}{x^{2} - 9}}$ equal to $0$ to find where the expression is undefined.

$\frac{x^{2} - 36}{x^{2} - 9} = 0$

Step 6

Solve for $x$ .

Set the numerator equal to zero.

$x^{2} - 36 = 0$

Solve the equation for $x$ .

Add $36$ to both sides of the equation.

$x^{2} = 36$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{36}$

Simplify $\pm \sqrt{36}$ .

Rewrite $36$ as $6^{2}$ .

$x = \pm \sqrt{6^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 6$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = 6$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 6$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 6, - 6$

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 $0$ .

$x = - 6, x = - 3, x = 3, x = 6$

Step 8

Find the Excluded Values ((x^2-9x+18)/(x^2+9x+18))÷((x^2-36)/(x^2-9))

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