Find Where Undefined/Discontinuous (x^2+7x+10)/(x+3)*(x^2-3x-18)/(x^2+x-2)

$\frac{x^{2} + 7 x + 10}{x + 3} \cdot \frac{x^{2} - 3 x - 18}{x^{2} + x - 2}$

Step 1

Set the denominator in $\frac{x^{2} + 7 x + 10}{x + 3}$ equal to $0$ to find where the expression is undefined.

$x + 3 = 0$

Step 2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 3

Set the denominator in $\frac{x^{2} - 3 x - 18}{x^{2} + x - 2}$ equal to $0$ to find where the expression is undefined.

$x^{2} + x - 2 = 0$

Step 4

Solve for $x$ .

Factor $x^{2} + x - 2$ 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 $- 2$ and whose sum is $1$ .

$- 1, 2$

Write the factored form using these integers.

$(x - 1) (x + 2) = 0$

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

$x - 1 = 0$

$x + 2 = 0$

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

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

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

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 $(x - 1) (x + 2) = 0$ true.

$x = 1, - 2$

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 $0$ .

$x = - 3, x = - 2, x = 1$

Step 6

Find Where Undefined/Discontinuous (x^2+7x+10)/(x+3)*(x^2-3x-18)/(x^2+x-2)

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