Calculus Examples

$f (x) = (x - 9) (x^{2} - 18 x - 162)$

Step 1

Find the x-intercepts.

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Step 1.1

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

$0 = (x - 9) (x^{2} - 18 x - 162)$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $(x - 9) (x^{2} - 18 x - 162) = 0$ .

$(x - 9) (x^{2} - 18 x - 162) = 0$

Step 1.2.2

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

$x - 9 = 0$

$x^{2} - 18 x - 162 = 0$

Step 1.2.3

Set $x - 9$ equal to $0$ and solve for $x$ .

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Step 1.2.3.1

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Step 1.2.3.2

Add $9$ to both sides of the equation.

$x = 9$

Step 1.2.4

Set $x^{2} - 18 x - 162$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $x^{2} - 18 x - 162$ equal to $0$ .

$x^{2} - 18 x - 162 = 0$

Step 1.2.4.2

Solve $x^{2} - 18 x - 162 = 0$ for $x$ .

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Step 1.2.4.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 1.2.4.2.2

Substitute the values $a = 1$ , $b = - 18$ , and $c = - 162$ into the quadratic formula and solve for $x$ .

$\frac{18 \pm \sqrt{{(- 18)}^{2} - 4 \cdot (1 \cdot - 162)}}{2 \cdot 1}$

Step 1.2.4.2.3

Simplify.

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Step 1.2.4.2.3.1

Simplify the numerator.

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Step 1.2.4.2.3.1.1

Raise $- 18$ to the power of $2$ .

$x = \frac{18 \pm \sqrt{324 - 4 \cdot 1 \cdot - 162}}{2 \cdot 1}$

Step 1.2.4.2.3.1.2

Multiply $- 4 \cdot 1 \cdot - 162$ .

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Step 1.2.4.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{18 \pm \sqrt{324 - 4 \cdot - 162}}{2 \cdot 1}$

Step 1.2.4.2.3.1.2.2

Multiply $- 4$ by $- 162$ .

$x = \frac{18 \pm \sqrt{324 + 648}}{2 \cdot 1}$

Step 1.2.4.2.3.1.3

Add $324$ and $648$ .

$x = \frac{18 \pm \sqrt{972}}{2 \cdot 1}$

Step 1.2.4.2.3.1.4

Rewrite $972$ as $18^{2} \cdot 3$ .

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Step 1.2.4.2.3.1.4.1

Factor $324$ out of $972$ .

$x = \frac{18 \pm \sqrt{324 (3)}}{2 \cdot 1}$

Step 1.2.4.2.3.1.4.2

Rewrite $324$ as $18^{2}$ .

$x = \frac{18 \pm \sqrt{18^{2} \cdot 3}}{2 \cdot 1}$

Step 1.2.4.2.3.1.5

Pull terms out from under the radical.

$x = \frac{18 \pm 18 \sqrt{3}}{2 \cdot 1}$

Step 1.2.4.2.3.2

Multiply $2$ by $1$ .

$x = \frac{18 \pm 18 \sqrt{3}}{2}$

Step 1.2.4.2.3.3

Simplify $\frac{18 \pm 18 \sqrt{3}}{2}$ .

$x = 9 \pm 9 \sqrt{3}$

Step 1.2.4.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 1.2.4.2.4.1

Simplify the numerator.

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Step 1.2.4.2.4.1.1

Raise $- 18$ to the power of $2$ .

$x = \frac{18 \pm \sqrt{324 - 4 \cdot 1 \cdot - 162}}{2 \cdot 1}$

Step 1.2.4.2.4.1.2

Multiply $- 4 \cdot 1 \cdot - 162$ .

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Step 1.2.4.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{18 \pm \sqrt{324 - 4 \cdot - 162}}{2 \cdot 1}$

Step 1.2.4.2.4.1.2.2

Multiply $- 4$ by $- 162$ .

$x = \frac{18 \pm \sqrt{324 + 648}}{2 \cdot 1}$

Step 1.2.4.2.4.1.3

Add $324$ and $648$ .

$x = \frac{18 \pm \sqrt{972}}{2 \cdot 1}$

Step 1.2.4.2.4.1.4

Rewrite $972$ as $18^{2} \cdot 3$ .

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Step 1.2.4.2.4.1.4.1

Factor $324$ out of $972$ .

$x = \frac{18 \pm \sqrt{324 (3)}}{2 \cdot 1}$

Step 1.2.4.2.4.1.4.2

Rewrite $324$ as $18^{2}$ .

$x = \frac{18 \pm \sqrt{18^{2} \cdot 3}}{2 \cdot 1}$

Step 1.2.4.2.4.1.5

Pull terms out from under the radical.

$x = \frac{18 \pm 18 \sqrt{3}}{2 \cdot 1}$

Step 1.2.4.2.4.2

Multiply $2$ by $1$ .

$x = \frac{18 \pm 18 \sqrt{3}}{2}$

Step 1.2.4.2.4.3

Simplify $\frac{18 \pm 18 \sqrt{3}}{2}$ .

$x = 9 \pm 9 \sqrt{3}$

Step 1.2.4.2.4.4

Change the $\pm$ to $+$ .

$x = 9 + 9 \sqrt{3}$

Step 1.2.4.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 1.2.4.2.5.1

Simplify the numerator.

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Step 1.2.4.2.5.1.1

Raise $- 18$ to the power of $2$ .

$x = \frac{18 \pm \sqrt{324 - 4 \cdot 1 \cdot - 162}}{2 \cdot 1}$

Step 1.2.4.2.5.1.2

Multiply $- 4 \cdot 1 \cdot - 162$ .

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Step 1.2.4.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{18 \pm \sqrt{324 - 4 \cdot - 162}}{2 \cdot 1}$

Step 1.2.4.2.5.1.2.2

Multiply $- 4$ by $- 162$ .

$x = \frac{18 \pm \sqrt{324 + 648}}{2 \cdot 1}$

Step 1.2.4.2.5.1.3

Add $324$ and $648$ .

$x = \frac{18 \pm \sqrt{972}}{2 \cdot 1}$

Step 1.2.4.2.5.1.4

Rewrite $972$ as $18^{2} \cdot 3$ .

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Step 1.2.4.2.5.1.4.1

Factor $324$ out of $972$ .

$x = \frac{18 \pm \sqrt{324 (3)}}{2 \cdot 1}$

Step 1.2.4.2.5.1.4.2

Rewrite $324$ as $18^{2}$ .

$x = \frac{18 \pm \sqrt{18^{2} \cdot 3}}{2 \cdot 1}$

Step 1.2.4.2.5.1.5

Pull terms out from under the radical.

$x = \frac{18 \pm 18 \sqrt{3}}{2 \cdot 1}$

Step 1.2.4.2.5.2

Multiply $2$ by $1$ .

$x = \frac{18 \pm 18 \sqrt{3}}{2}$

Step 1.2.4.2.5.3

Simplify $\frac{18 \pm 18 \sqrt{3}}{2}$ .

$x = 9 \pm 9 \sqrt{3}$

Step 1.2.4.2.5.4

Change the $\pm$ to $-$ .

$x = 9 - 9 \sqrt{3}$

Step 1.2.4.2.6

The final answer is the combination of both solutions.

$x = 9 + 9 \sqrt{3}, 9 - 9 \sqrt{3}$

Step 1.2.5

The final solution is all the values that make $(x - 9) (x^{2} - 18 x - 162) = 0$ true.

$x = 9, 9 + 9 \sqrt{3}, 9 - 9 \sqrt{3}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(9, 0), (9 + 9 \sqrt{3}, 0), (9 - 9 \sqrt{3}, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

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

$y = ((0) - 9) ({(0)}^{2} - 18 \cdot 0 - 162)$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = (0 - 9) ({(0)}^{2} - 18 \cdot 0 - 162)$

Step 2.2.2

Remove parentheses.

$y = (0 - 9) (0^{2} - 18 \cdot 0 - 162)$

Step 2.2.3

Remove parentheses.

$y = ((0) - 9) ({(0)}^{2} - 18 \cdot 0 - 162)$

Step 2.2.4

Simplify $((0) - 9) ({(0)}^{2} - 18 \cdot 0 - 162)$ .

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Step 2.2.4.1

Subtract $9$ from $0$ .

$y = - 9 ({(0)}^{2} - 18 \cdot 0 - 162)$

Step 2.2.4.2

Simplify each term.

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Step 2.2.4.2.1

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

$y = - 9 (0 - 18 \cdot 0 - 162)$

Step 2.2.4.2.2

Multiply $- 18$ by $0$ .

$y = - 9 (0 + 0 - 162)$

Step 2.2.4.3

Simplify the expression.

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Step 2.2.4.3.1

Add $0$ and $0$ .

$y = - 9 (0 - 162)$

Step 2.2.4.3.2

Subtract $162$ from $0$ .

$y = - 9 \cdot - 162$

Step 2.2.4.3.3

Multiply $- 9$ by $- 162$ .

$y = 1458$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 1458)$

Step 3

List the intersections.

x-intercept(s): $(9, 0), (9 + 9 \sqrt{3}, 0), (9 - 9 \sqrt{3}, 0)$

y-intercept(s): $(0, 1458)$

Step 4