Finite Math Examples

$f (x) = 4 x^{3} - 5 x^{2} - 2 x$

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 $x$ .

$0 = 4 x^{3} - 5 x^{2} - 2 x$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $4 x^{3} - 5 x^{2} - 2 x = 0$ .

$4 x^{3} - 5 x^{2} - 2 x = 0$

Step 1.2.2

Factor $x$ out of $4 x^{3} - 5 x^{2} - 2 x$ .

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Step 1.2.2.1

Factor $x$ out of $4 x^{3}$ .

$x (4 x^{2}) - 5 x^{2} - 2 x = 0$

Step 1.2.2.2

Factor $x$ out of $- 5 x^{2}$ .

$x (4 x^{2}) + x (- 5 x) - 2 x = 0$

Step 1.2.2.3

Factor $x$ out of $- 2 x$ .

$x (4 x^{2}) + x (- 5 x) + x \cdot - 2 = 0$

Step 1.2.2.4

Factor $x$ out of $x (4 x^{2}) + x (- 5 x)$ .

$x (4 x^{2} - 5 x) + x \cdot - 2 = 0$

Step 1.2.2.5

Factor $x$ out of $x (4 x^{2} - 5 x) + x \cdot - 2$ .

$x (4 x^{2} - 5 x - 2) = 0$

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 $0$ .

$x = 0$

$4 x^{2} - 5 x - 2 = 0$

Step 1.2.4

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.5

Set $4 x^{2} - 5 x - 2$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $4 x^{2} - 5 x - 2$ equal to $0$ .

$4 x^{2} - 5 x - 2 = 0$

Step 1.2.5.2

Solve $4 x^{2} - 5 x - 2 = 0$ 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 $a = 4$ , $b = - 5$ , and $c = - 2$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (4 \cdot - 2)}}{2 \cdot 4}$

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 $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 4 \cdot - 2}}{2 \cdot 4}$

Step 1.2.5.2.3.1.2

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

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Step 1.2.5.2.3.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{5 \pm \sqrt{25 - 16 \cdot - 2}}{2 \cdot 4}$

Step 1.2.5.2.3.1.2.2

Multiply $- 16$ by $- 2$ .

$x = \frac{5 \pm \sqrt{25 + 32}}{2 \cdot 4}$

Step 1.2.5.2.3.1.3

Add $25$ and $32$ .

$x = \frac{5 \pm \sqrt{57}}{2 \cdot 4}$

Step 1.2.5.2.3.2

Multiply $2$ by $4$ .

$x = \frac{5 \pm \sqrt{57}}{8}$

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 $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 4 \cdot - 2}}{2 \cdot 4}$

Step 1.2.5.2.4.1.2

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

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Step 1.2.5.2.4.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{5 \pm \sqrt{25 - 16 \cdot - 2}}{2 \cdot 4}$

Step 1.2.5.2.4.1.2.2

Multiply $- 16$ by $- 2$ .

$x = \frac{5 \pm \sqrt{25 + 32}}{2 \cdot 4}$

Step 1.2.5.2.4.1.3

Add $25$ and $32$ .

$x = \frac{5 \pm \sqrt{57}}{2 \cdot 4}$

Step 1.2.5.2.4.2

Multiply $2$ by $4$ .

$x = \frac{5 \pm \sqrt{57}}{8}$

Step 1.2.5.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{5 + \sqrt{57}}{8}$

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 $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 4 \cdot - 2}}{2 \cdot 4}$

Step 1.2.5.2.5.1.2

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

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Step 1.2.5.2.5.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{5 \pm \sqrt{25 - 16 \cdot - 2}}{2 \cdot 4}$

Step 1.2.5.2.5.1.2.2

Multiply $- 16$ by $- 2$ .

$x = \frac{5 \pm \sqrt{25 + 32}}{2 \cdot 4}$

Step 1.2.5.2.5.1.3

Add $25$ and $32$ .

$x = \frac{5 \pm \sqrt{57}}{2 \cdot 4}$

Step 1.2.5.2.5.2

Multiply $2$ by $4$ .

$x = \frac{5 \pm \sqrt{57}}{8}$

Step 1.2.5.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{5 - \sqrt{57}}{8}$

Step 1.2.5.2.6

The final answer is the combination of both solutions.

$x = \frac{5 + \sqrt{57}}{8}, \frac{5 - \sqrt{57}}{8}$

Step 1.2.6

The final solution is all the values that make $x (4 x^{2} - 5 x - 2) = 0$ true.

$x = 0, \frac{5 + \sqrt{57}}{8}, \frac{5 - \sqrt{57}}{8}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (\frac{5 + \sqrt{57}}{8}, 0), (\frac{5 - \sqrt{57}}{8}, 0)$

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 $y$ .

$y = 4 {(0)}^{3} - 5 {(0)}^{2} - 2 \cdot 0$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 4 \cdot 0^{3} - 5 {(0)}^{2} - 2 \cdot 0$

Step 2.2.2

Remove parentheses.

$y = 4 \cdot 0^{3} - 5 \cdot 0^{2} - 2 \cdot 0$

Step 2.2.3

Remove parentheses.

$y = 4 {(0)}^{3} - 5 {(0)}^{2} - 2 \cdot 0$

Step 2.2.4

Simplify $4 {(0)}^{3} - 5 {(0)}^{2} - 2 \cdot 0$ .

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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 $0$ .

$y = 4 \cdot 0 - 5 {(0)}^{2} - 2 \cdot 0$

Step 2.2.4.1.2

Multiply $4$ by $0$ .

$y = 0 - 5 {(0)}^{2} - 2 \cdot 0$

Step 2.2.4.1.3

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

$y = 0 - 5 \cdot 0 - 2 \cdot 0$

Step 2.2.4.1.4

Multiply $- 5$ by $0$ .

$y = 0 + 0 - 2 \cdot 0$

Step 2.2.4.1.5

Multiply $- 2$ 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): $(0, 0), (\frac{5 + \sqrt{57}}{8}, 0), (\frac{5 - \sqrt{57}}{8}, 0)$

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

Step 4