Precalculus Examples

$\frac{x^{2} + x - 72}{x^{2} - x - 56}$

Step 1

Set the denominator in $\frac{x^{2} + x - 72}{x^{2} - x - 56}$ equal to $0$ to find where the expression is undefined.

$x^{2} - x - 56 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor $x^{2} - x - 56$ using the AC method.

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Step 2.1.1

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 $- 56$ and whose sum is $- 1$ .

$- 8, 7$

Step 2.1.2

Write the factored form using these integers.

$(x - 8) (x + 7) = 0$

Step 2.2

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

$x - 8 = 0$

$x + 7 = 0$

Step 2.3

Set $x - 8$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Step 2.3.2

Add $8$ to both sides of the equation.

$x = 8$

Step 2.4

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

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Step 2.4.1

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

$x + 7 = 0$

Step 2.4.2

Subtract $7$ from both sides of the equation.

$x = - 7$

Step 2.5

The final solution is all the values that make $(x - 8) (x + 7) = 0$ true.

$x = 8, - 7$

Step 3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, - 7) \cup (- 7, 8) \cup (8, \infty)$

Set-Builder Notation:

${x | x \neq - 7, 8}$

Step 4