Precalculus Examples

$t (x) = \frac{x^{2} - 3 x - 18}{x - 9}$

Step 1

Find the x-intercepts.

Tap for more steps...

Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = \frac{x^{2} - 3 x - 18}{x - 9}$

Step 1.2

Solve the equation.

Tap for more steps...

Step 1.2.1

Set the numerator equal to zero.

$x^{2} - 3 x - 18 = 0$

Step 1.2.2

Solve the equation for $x$ .

Tap for more steps...

Step 1.2.2.1

Factor $x^{2} - 3 x - 18$ using the AC method.

Tap for more steps...

Step 1.2.2.1.1

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 $- 18$ and whose sum is $- 3$ .

$- 6, 3$

Step 1.2.2.1.2

Write the factored form using these integers.

$(x - 6) (x + 3) = 0$

Step 1.2.2.2

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

$x - 6 = 0$

$x + 3 = 0$

Step 1.2.2.3

Set $x - 6$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 1.2.2.3.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 1.2.2.3.2

Add $6$ to both sides of the equation.

$x = 6$

Step 1.2.2.4

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

Tap for more steps...

Step 1.2.2.4.1

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

$x + 3 = 0$

Step 1.2.2.4.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 1.2.2.5

The final solution is all the values that make $(x - 6) (x + 3) = 0$ true.

$x = 6, - 3$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(6, 0), (- 3, 0)$

Step 2

Find the y-intercepts.

Tap for more steps...

Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = \frac{{(0)}^{2} - 3 \cdot 0 - 18}{(0) - 9}$

Step 2.2

Solve the equation.

Tap for more steps...

Step 2.2.1

Remove parentheses.

$y = \frac{0^{2} - 3 \cdot 0 - 18}{(0) - 9}$

Step 2.2.2

Remove parentheses.

$y = \frac{0^{2} - 3 \cdot 0 - 18}{0 - 9}$

Step 2.2.3

Remove parentheses.

$y = \frac{{(0)}^{2} - 3 \cdot 0 - 18}{(0) - 9}$

Step 2.2.4

Simplify $\frac{{(0)}^{2} - 3 \cdot 0 - 18}{(0) - 9}$ .

Tap for more steps...

Step 2.2.4.1

Simplify the numerator.

Tap for more steps...

Step 2.2.4.1.1

Raising $0$ to any positive power yields $0$ .

$y = \frac{0 - 3 \cdot 0 - 18}{0 - 9}$

Step 2.2.4.1.2

Multiply $- 3$ by $0$ .

$y = \frac{0 + 0 - 18}{0 - 9}$

Step 2.2.4.1.3

Add $0$ and $0$ .

$y = \frac{0 - 18}{0 - 9}$

Step 2.2.4.1.4

Subtract $18$ from $0$ .

$y = \frac{- 18}{0 - 9}$

Step 2.2.4.2

Simplify the expression.

Tap for more steps...

Step 2.2.4.2.1

Subtract $9$ from $0$ .

$y = \frac{- 18}{- 9}$

Step 2.2.4.2.2

Divide $- 18$ by $- 9$ .

$y = 2$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 2)$

Step 3

List the intersections.

x-intercept(s): $(6, 0), (- 3, 0)$

y-intercept(s): $(0, 2)$

Step 4