Finite Math Examples

$f (x) = 2 x^{3} - 15 x^{2} - 144 x$

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 $x$ .

$0 = 2 x^{3} - 15 x^{2} - 144 x$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $2 x^{3} - 15 x^{2} - 144 x = 0$ .

$2 x^{3} - 15 x^{2} - 144 x = 0$

Step 1.2.2

Factor $x$ out of $2 x^{3} - 15 x^{2} - 144 x$ .

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Step 1.2.2.1

Factor $x$ out of $2 x^{3}$ .

$x (2 x^{2}) - 15 x^{2} - 144 x = 0$

Step 1.2.2.2

Factor $x$ out of $- 15 x^{2}$ .

$x (2 x^{2}) + x (- 15 x) - 144 x = 0$

Step 1.2.2.3

Factor $x$ out of $- 144 x$ .

$x (2 x^{2}) + x (- 15 x) + x \cdot - 144 = 0$

Step 1.2.2.4

Factor $x$ out of $x (2 x^{2}) + x (- 15 x)$ .

$x (2 x^{2} - 15 x) + x \cdot - 144 = 0$

Step 1.2.2.5

Factor $x$ out of $x (2 x^{2} - 15 x) + x \cdot - 144$ .

$x (2 x^{2} - 15 x - 144) = 0$

Step 1.2.3

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

$x = 0$

$2 x^{2} - 15 x - 144 = 0$

Step 1.2.4

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.5

Set $2 x^{2} - 15 x - 144$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $2 x^{2} - 15 x - 144$ equal to $0$ .

$2 x^{2} - 15 x - 144 = 0$

Step 1.2.5.2

Solve $2 x^{2} - 15 x - 144 = 0$ for $x$ .

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Step 1.2.5.2.1

Use the quadratic formula to find the solutions.

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

Step 1.2.5.2.2

Substitute the values $a = 2$ , $b = - 15$ , and $c = - 144$ into the quadratic formula and solve for $x$ .

$\frac{15 \pm \sqrt{{(- 15)}^{2} - 4 \cdot (2 \cdot - 144)}}{2 \cdot 2}$

Step 1.2.5.2.3

Simplify.

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Step 1.2.5.2.3.1

Simplify the numerator.

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Step 1.2.5.2.3.1.1

Raise $- 15$ to the power of $2$ .

$x = \frac{15 \pm \sqrt{225 - 4 \cdot 2 \cdot - 144}}{2 \cdot 2}$

Step 1.2.5.2.3.1.2

Multiply $- 4 \cdot 2 \cdot - 144$ .

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Step 1.2.5.2.3.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{15 \pm \sqrt{225 - 8 \cdot - 144}}{2 \cdot 2}$

Step 1.2.5.2.3.1.2.2

Multiply $- 8$ by $- 144$ .

$x = \frac{15 \pm \sqrt{225 + 1152}}{2 \cdot 2}$

Step 1.2.5.2.3.1.3

Add $225$ and $1152$ .

$x = \frac{15 \pm \sqrt{1377}}{2 \cdot 2}$

Step 1.2.5.2.3.1.4

Rewrite $1377$ as $9^{2} \cdot 17$ .

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Step 1.2.5.2.3.1.4.1

Factor $81$ out of $1377$ .

$x = \frac{15 \pm \sqrt{81 (17)}}{2 \cdot 2}$

Step 1.2.5.2.3.1.4.2

Rewrite $81$ as $9^{2}$ .

$x = \frac{15 \pm \sqrt{9^{2} \cdot 17}}{2 \cdot 2}$

Step 1.2.5.2.3.1.5

Pull terms out from under the radical.

$x = \frac{15 \pm 9 \sqrt{17}}{2 \cdot 2}$

Step 1.2.5.2.3.2

Multiply $2$ by $2$ .

$x = \frac{15 \pm 9 \sqrt{17}}{4}$

Step 1.2.5.2.4

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

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Step 1.2.5.2.4.1

Simplify the numerator.

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Step 1.2.5.2.4.1.1

Raise $- 15$ to the power of $2$ .

$x = \frac{15 \pm \sqrt{225 - 4 \cdot 2 \cdot - 144}}{2 \cdot 2}$

Step 1.2.5.2.4.1.2

Multiply $- 4 \cdot 2 \cdot - 144$ .

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Step 1.2.5.2.4.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{15 \pm \sqrt{225 - 8 \cdot - 144}}{2 \cdot 2}$

Step 1.2.5.2.4.1.2.2

Multiply $- 8$ by $- 144$ .

$x = \frac{15 \pm \sqrt{225 + 1152}}{2 \cdot 2}$

Step 1.2.5.2.4.1.3

Add $225$ and $1152$ .

$x = \frac{15 \pm \sqrt{1377}}{2 \cdot 2}$

Step 1.2.5.2.4.1.4

Rewrite $1377$ as $9^{2} \cdot 17$ .

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Step 1.2.5.2.4.1.4.1

Factor $81$ out of $1377$ .

$x = \frac{15 \pm \sqrt{81 (17)}}{2 \cdot 2}$

Step 1.2.5.2.4.1.4.2

Rewrite $81$ as $9^{2}$ .

$x = \frac{15 \pm \sqrt{9^{2} \cdot 17}}{2 \cdot 2}$

Step 1.2.5.2.4.1.5

Pull terms out from under the radical.

$x = \frac{15 \pm 9 \sqrt{17}}{2 \cdot 2}$

Step 1.2.5.2.4.2

Multiply $2$ by $2$ .

$x = \frac{15 \pm 9 \sqrt{17}}{4}$

Step 1.2.5.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{15 + 9 \sqrt{17}}{4}$

Step 1.2.5.2.5

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

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Step 1.2.5.2.5.1

Simplify the numerator.

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Step 1.2.5.2.5.1.1

Raise $- 15$ to the power of $2$ .

$x = \frac{15 \pm \sqrt{225 - 4 \cdot 2 \cdot - 144}}{2 \cdot 2}$

Step 1.2.5.2.5.1.2

Multiply $- 4 \cdot 2 \cdot - 144$ .

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Step 1.2.5.2.5.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{15 \pm \sqrt{225 - 8 \cdot - 144}}{2 \cdot 2}$

Step 1.2.5.2.5.1.2.2

Multiply $- 8$ by $- 144$ .

$x = \frac{15 \pm \sqrt{225 + 1152}}{2 \cdot 2}$

Step 1.2.5.2.5.1.3

Add $225$ and $1152$ .

$x = \frac{15 \pm \sqrt{1377}}{2 \cdot 2}$

Step 1.2.5.2.5.1.4

Rewrite $1377$ as $9^{2} \cdot 17$ .

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Step 1.2.5.2.5.1.4.1

Factor $81$ out of $1377$ .

$x = \frac{15 \pm \sqrt{81 (17)}}{2 \cdot 2}$

Step 1.2.5.2.5.1.4.2

Rewrite $81$ as $9^{2}$ .

$x = \frac{15 \pm \sqrt{9^{2} \cdot 17}}{2 \cdot 2}$

Step 1.2.5.2.5.1.5

Pull terms out from under the radical.

$x = \frac{15 \pm 9 \sqrt{17}}{2 \cdot 2}$

Step 1.2.5.2.5.2

Multiply $2$ by $2$ .

$x = \frac{15 \pm 9 \sqrt{17}}{4}$

Step 1.2.5.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{15 - 9 \sqrt{17}}{4}$

Step 1.2.5.2.6

The final answer is the combination of both solutions.

$x = \frac{15 + 9 \sqrt{17}}{4}, \frac{15 - 9 \sqrt{17}}{4}$

Step 1.2.6

The final solution is all the values that make $x (2 x^{2} - 15 x - 144) = 0$ true.

$x = 0, \frac{15 + 9 \sqrt{17}}{4}, \frac{15 - 9 \sqrt{17}}{4}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (\frac{15 + 9 \sqrt{17}}{4}, 0), (\frac{15 - 9 \sqrt{17}}{4}, 0)$

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 $y$ .

$y = 2 {(0)}^{3} - 15 {(0)}^{2} - 144 \cdot 0$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 2 \cdot 0^{3} - 15 {(0)}^{2} - 144 \cdot 0$

Step 2.2.2

Remove parentheses.

$y = 2 \cdot 0^{3} - 15 \cdot 0^{2} - 144 \cdot 0$

Step 2.2.3

Remove parentheses.

$y = 2 {(0)}^{3} - 15 {(0)}^{2} - 144 \cdot 0$

Step 2.2.4

Simplify $2 {(0)}^{3} - 15 {(0)}^{2} - 144 \cdot 0$ .

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Step 2.2.4.1

Simplify each term.

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Step 2.2.4.1.1

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

$y = 2 \cdot 0 - 15 {(0)}^{2} - 144 \cdot 0$

Step 2.2.4.1.2

Multiply $2$ by $0$ .

$y = 0 - 15 {(0)}^{2} - 144 \cdot 0$

Step 2.2.4.1.3

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

$y = 0 - 15 \cdot 0 - 144 \cdot 0$

Step 2.2.4.1.4

Multiply $- 15$ by $0$ .

$y = 0 + 0 - 144 \cdot 0$

Step 2.2.4.1.5

Multiply $- 144$ by $0$ .

$y = 0 + 0 + 0$

Step 2.2.4.2

Simplify by adding numbers.

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Step 2.2.4.2.1

Add $0$ and $0$ .

$y = 0 + 0$

Step 2.2.4.2.2

Add $0$ and $0$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (\frac{15 + 9 \sqrt{17}}{4}, 0), (\frac{15 - 9 \sqrt{17}}{4}, 0)$

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

Step 4