Find Where Undefined/Discontinuous (x^2-11x+28)/(x^2-16)*(x^2+7x+12)/(x^2-4x-21)

$\frac{x^{2} - 11 x + 28}{x^{2} - 16} \cdot \frac{x^{2} + 7 x + 12}{x^{2} - 4 x - 21}$

Step 1

Set the denominator in $\frac{x^{2} - 11 x + 28}{x^{2} - 16}$ 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 $\frac{x^{2} + 7 x + 12}{x^{2} - 4 x - 21}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 4 x - 21 = 0$

Step 4

Solve for $x$ .

Factor $x^{2} - 4 x - 21$ 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 $- 21$ and whose sum is $- 4$ .

$- 7, 3$

Write the factored form using these integers.

$(x - 7) (x + 3) = 0$

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

$x - 7 = 0$

$x + 3 = 0$

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 $x + 3$ equal to $0$ and solve for $x$ .

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

The final solution is all the values that make $(x - 7) (x + 3) = 0$ true.

$x = 7, - 3$

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 $0$ .

$x = - 4, x = - 3, x = 4, x = 7$

Step 6

Find Where Undefined/Discontinuous (x^2-11x+28)/(x^2-16)*(x^2+7x+12)/(x^2-4x-21)

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