Find Where Undefined/Discontinuous (x^2-9x+20)/(x^2-16)*(x^2+6x+8)/(x^2-3x-10)

$\frac{x^{2} - 9 x + 20}{x^{2} - 16} \cdot \frac{x^{2} + 6 x + 8}{x^{2} - 3 x - 10}$

Step 1

Set the denominator in $\frac{x^{2} - 9 x + 20}{x^{2} - 16}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 16 = 0$

Step 2

Solve for $x$ .

Add $16$ to both sides of the equation.

$x^{2} = 16$

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

$x = \pm \sqrt{16}$

Simplify $\pm \sqrt{16}$ .

Rewrite $16$ as $4^{2}$ .

$x = \pm \sqrt{4^{2}}$

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

$x = \pm 4$

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.

$x = 4$

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

$x = - 4$

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

$x = 4, - 4$

Step 3

Set the denominator in $\frac{x^{2} + 6 x + 8}{x^{2} - 3 x - 10}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 3 x - 10 = 0$

Step 4

Solve for $x$ .

Factor $x^{2} - 3 x - 10$ 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 $- 10$ and whose sum is $- 3$ .

$- 5, 2$

Write the factored form using these integers.

$(x - 5) (x + 2) = 0$

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

$x - 5 = 0$

$x + 2 = 0$

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

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 $(x - 5) (x + 2) = 0$ true.

$x = 5, - 2$

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 $0$ .

$x = - 4, x = - 2, x = 4, x = 5$

Step 6

Find Where Undefined/Discontinuous (x^2-9x+20)/(x^2-16)*(x^2+6x+8)/(x^2-3x-10)

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