Algebra Examples

$f (x) = \frac{x^{2} - 100}{x^{2} - 7 x - 30}$

Step 1

Set the denominator in $\frac{x^{2} - 100}{x^{2} - 7 x - 30}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 7 x - 30 = 0$

Step 2

Solve for $x$ .

Tap for more steps...

Factor $x^{2} - 7 x - 30$ using the AC method.

Tap for more steps...

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 $- 30$ and whose sum is $- 7$ .

$- 10, 3$

Write the factored form using these integers.

$(x - 10) (x + 3) = 0$

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

$x - 10 = 0$

$x + 3 = 0$

Set $x - 10$ equal to $0$ and solve for $x$ .

Tap for more steps...

Set $x - 10$ equal to $0$ .

$x - 10 = 0$

Add $10$ to both sides of the equation.

$x = 10$

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

Tap for more steps...

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 $(x - 10) (x + 3) = 0$ true.

$x = 10, - 3$

Step 3

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 $0$ .

$x = - 3, x = 10$

Step 4