Find Where Undefined/Discontinuous f(x)=(x^2-2x-24)/(x^2+10x+24)

$f (x) = \frac{x^{2} - 2 x - 24}{x^{2} + 10 x + 24}$

Step 1

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

$x^{2} + 10 x + 24 = 0$

Step 2

Solve for $x$ .

Factor $x^{2} + 10 x + 24$ 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 $10$ .

$4, 6$

Write the factored form using these integers.

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

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

$x + 4 = 0$

$x + 6 = 0$

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.

$x = - 4$

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 $(x + 4) (x + 6) = 0$ true.

$x = - 4, - 6$

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 $0$ .

$x = - 6, x = - 4$

Step 4

Find Where Undefined/Discontinuous f(x)=(x^2-2x-24)/(x^2+10x+24)

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