Find the Excluded Values (x+6)/(x^2-7x-18)-(2x)/(x-9)

$\frac{x + 6}{x^{2} - 7 x - 18} - \frac{2 x}{x - 9}$

Step 1

Set the denominator in $\frac{x + 6}{x^{2} - 7 x - 18}$ equal to $0$ to find where the expression is undefined.

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

Step 2

Solve 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.

$x = 9, - 2$

Step 3

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

$x - 9 = 0$

Step 4

Add $9$ to both sides of the equation.

$x = 9$

Step 5

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 $0$ .

$x = - 2, x = 9$

Step 6

Find the Excluded Values (x+6)/(x^2-7x-18)-(2x)/(x-9)

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