Find Where Undefined/Discontinuous (x^2-11x+30)/(x^2-25)*(x^2+7x+10)/(x^2-4x-12)

$\frac{x^{2} - 11 x + 30}{x^{2} - 25} \cdot \frac{x^{2} + 7 x + 10}{x^{2} - 4 x - 12}$

Step 1

Set the denominator in $\frac{x^{2} - 11 x + 30}{x^{2} - 25}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 25 = 0$

Step 2

Solve for $x$ .

Add $25$ to both sides of the equation.

$x^{2} = 25$

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

$x = \pm \sqrt{25}$

Simplify $\pm \sqrt{25}$ .

Rewrite $25$ as $5^{2}$ .

$x = \pm \sqrt{5^{2}}$

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

$x = \pm 5$

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.

$x = 5$

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

$x = - 5$

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

$x = 5, - 5$

Step 3

Set the denominator in $\frac{x^{2} + 7 x + 10}{x^{2} - 4 x - 12}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 4 x - 12 = 0$

Step 4

Solve for $x$ .

Factor $x^{2} - 4 x - 12$ 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 $- 12$ and whose sum is $- 4$ .

$- 6, 2$

Write the factored form using these integers.

$(x - 6) (x + 2) = 0$

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

$x - 6 = 0$

$x + 2 = 0$

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

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

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

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 $(x - 6) (x + 2) = 0$ true.

$x = 6, - 2$

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 $0$ .

$x = - 5, x = - 2, x = 5, x = 6$

Step 6

Find Where Undefined/Discontinuous (x^2-11x+30)/(x^2-25)*(x^2+7x+10)/(x^2-4x-12)

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