Find Where Undefined/Discontinuous (x^2-10x+21)/(x^2-9)*(x^2+5x+6)/(x^2-5x-14)

$\frac{x^{2} - 10 x + 21}{x^{2} - 9} \cdot \frac{x^{2} + 5 x + 6}{x^{2} - 5 x - 14}$

Step 1

Set the denominator in $\frac{x^{2} - 10 x + 21}{x^{2} - 9}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 9 = 0$

Step 2

Solve for $x$ .

Add $9$ to both sides of the equation.

$x^{2} = 9$

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

$x = \pm \sqrt{9}$

Simplify $\pm \sqrt{9}$ .

Rewrite $9$ as $3^{2}$ .

$x = \pm \sqrt{3^{2}}$

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

$x = \pm 3$

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.

$x = 3$

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

$x = - 3$

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

$x = 3, - 3$

Step 3

Set the denominator in $\frac{x^{2} + 5 x + 6}{x^{2} - 5 x - 14}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 5 x - 14 = 0$

Step 4

Solve for $x$ .

Factor $x^{2} - 5 x - 14$ 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 $- 14$ and whose sum is $- 5$ .

$- 7, 2$

Write the factored form using these integers.

$(x - 7) (x + 2) = 0$

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

$x - 7 = 0$

$x + 2 = 0$

Set $x - 7$ equal to $0$ and solve for $x$ .

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

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 $(x - 7) (x + 2) = 0$ true.

$x = 7, - 2$

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 $0$ .

$x = - 3, x = - 2, x = 3, x = 7$

Step 6

Find Where Undefined/Discontinuous (x^2-10x+21)/(x^2-9)*(x^2+5x+6)/(x^2-5x-14)

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