Find the Excluded Values ((x^2-10x+9)/(3x))÷((x^2-7x-18)/(x^2+2x))

$\frac{x^{2} - 10 x + 9}{3 x} \div \frac{x^{2} - 7 x - 18}{x^{2} + 2 x}$

Step 1

Set the denominator in $\frac{x^{2} - 10 x + 9}{3 x}$ equal to $0$ to find where the expression is undefined.

$3 x = 0$

Step 2

Divide each term in $3 x = 0$ by $3$ and simplify.

Divide each term in $3 x = 0$ by $3$ .

$\frac{3 x}{3} = \frac{0}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{0}{3}$

Divide $x$ by $1$ .

$x = \frac{0}{3}$

Simplify the right side.

Divide $0$ by $3$ .

$x = 0$

Step 3

Set the denominator in $\frac{x^{2} - 7 x - 18}{x^{2} + 2 x}$ equal to $0$ to find where the expression is undefined.

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

Step 4

Solve for $x$ .

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

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

$x \cdot x + 2 x = 0$

Factor $x$ out of $2 x$ .

$x \cdot x + x \cdot 2 = 0$

Factor $x$ out of $x \cdot x + x \cdot 2$ .

$x (x + 2) = 0$

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

$x = 0$

$x + 2 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x + 2$ equal to $0$ and solve for $x$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

The final solution is all the values that make $x (x + 2) = 0$ true.

$x = 0, - 2$

Step 5

Set the denominator in $\frac{\frac{x^{2} - 10 x + 9}{3 x}}{\frac{x^{2} - 7 x - 18}{x^{2} + 2 x}}$ equal to $0$ to find where the expression is undefined.

$\frac{x^{2} - 7 x - 18}{x^{2} + 2 x} = 0$

Step 6

Solve for $x$ .

Set the numerator equal to zero.

$x^{2} - 7 x - 18 = 0$

Solve the equation for $x$ .

Factor $x^{2} - 7 x - 18$ 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 $- 7$ .

$- 9, 2$

Write the factored form using these integers.

$(x - 9) (x + 2) = 0$

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

$x - 9 = 0$

$x + 2 = 0$

Set $x - 9$ equal to $0$ and solve for $x$ .

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Add $9$ to both sides of the equation.

$x = 9$

Set $x + 2$ equal to $0$ and solve for $x$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

The final solution is all the values that make $(x - 9) (x + 2) = 0$ true.

$x = 9, - 2$

Exclude the solutions that do not make $\frac{x^{2} - 7 x - 18}{x^{2} + 2 x} = 0$ true.

$x = 9$

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 $0$ .

$x = - 2, x = 0, x = 9$

Step 8

Find the Excluded Values ((x^2-10x+9)/(3x))÷((x^2-7x-18)/(x^2+2x))

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