Find Where Undefined/Discontinuous (x^2-10x-24)/(x^2-3x-108)

$\frac{(x^{2} - 10 x - 24)}{x^{2} - 3 x - 108}$

Step 1

Set the denominator in $\frac{x^{2} - 10 x - 24}{x^{2} - 3 x - 108}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 3 x - 108 = 0$

Step 2

Solve for $x$ .

Factor $x^{2} - 3 x - 108$ 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 $- 108$ and whose sum is $- 3$ .

$- 12, 9$

Write the factored form using these integers.

$(x - 12) (x + 9) = 0$

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

$x - 12 = 0$

$x + 9 = 0$

Set $x - 12$ equal to $0$ and solve for $x$ .

Set $x - 12$ equal to $0$ .

$x - 12 = 0$

Add $12$ to both sides of the equation.

$x = 12$

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

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

$x + 9 = 0$

Subtract $9$ from both sides of the equation.

$x = - 9$

The final solution is all the values that make $(x - 12) (x + 9) = 0$ true.

$x = 12, - 9$

Step 3

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 $0$ .

$x = - 9, x = 12$

Step 4

Find Where Undefined/Discontinuous (x^2-10x-24)/(x^2-3x-108)

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