Find Where Undefined/Discontinuous (x^2-13x+40)/(x^2-25)*(x^2+6x+5)/(x^2-7x-8)

$\frac{x^{2} - 13 x + 40}{x^{2} - 25} \cdot \frac{x^{2} + 6 x + 5}{x^{2} - 7 x - 8}$

Step 1

Set the denominator in $\frac{x^{2} - 13 x + 40}{x^{2} - 25}$ 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 $\frac{x^{2} + 6 x + 5}{x^{2} - 7 x - 8}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 7 x - 8 = 0$

Step 4

Solve for $x$ .

Factor $x^{2} - 7 x - 8$ 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 $- 8$ and whose sum is $- 7$ .

$- 8, 1$

Write the factored form using these integers.

$(x - 8) (x + 1) = 0$

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

$x - 8 = 0$

$x + 1 = 0$

Set $x - 8$ equal to $0$ and solve for $x$ .

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Add $8$ to both sides of the equation.

$x = 8$

Set $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

The final solution is all the values that make $(x - 8) (x + 1) = 0$ true.

$x = 8, - 1$

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 $0$ .

$x = - 5, x = - 1, x = 5, x = 8$

Step 6

Find Where Undefined/Discontinuous (x^2-13x+40)/(x^2-25)*(x^2+6x+5)/(x^2-7x-8)

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