Finite Math Examples

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

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

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

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

Use the quadratic formula to find the solutions.

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

Step 1.2.2.2

Substitute the values $a = 3$ , $b = 4$ , and $c = 5$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (3 \cdot 5)}}{2 \cdot 3}$

Step 1.2.2.3

Simplify.

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Step 1.2.2.3.1

Simplify the numerator.

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Step 1.2.2.3.1.1

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 3 \cdot 5}}{2 \cdot 3}$

Step 1.2.2.3.1.2

Multiply $- 4 \cdot 3 \cdot 5$ .

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Step 1.2.2.3.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{- 4 \pm \sqrt{16 - 12 \cdot 5}}{2 \cdot 3}$

Step 1.2.2.3.1.2.2

Multiply $- 12$ by $5$ .

$x = \frac{- 4 \pm \sqrt{16 - 60}}{2 \cdot 3}$

Step 1.2.2.3.1.3

Subtract $60$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 44}}{2 \cdot 3}$

Step 1.2.2.3.1.4

Rewrite $- 44$ as $- 1 (44)$ .

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 44}}{2 \cdot 3}$

Step 1.2.2.3.1.5

Rewrite $\sqrt{- 1 (44)}$ as $\sqrt{- 1} \cdot \sqrt{44}$ .

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{44}}{2 \cdot 3}$

Step 1.2.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 4 \pm i \cdot \sqrt{44}}{2 \cdot 3}$

Step 1.2.2.3.1.7

Rewrite $44$ as $2^{2} \cdot 11$ .

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Step 1.2.2.3.1.7.1

Factor $4$ out of $44$ .

$x = \frac{- 4 \pm i \cdot \sqrt{4 (11)}}{2 \cdot 3}$

Step 1.2.2.3.1.7.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 4 \pm i \cdot \sqrt{2^{2} \cdot 11}}{2 \cdot 3}$

Step 1.2.2.3.1.8

Pull terms out from under the radical.

$x = \frac{- 4 \pm i \cdot (2 \sqrt{11})}{2 \cdot 3}$

Step 1.2.2.3.1.9

Move $2$ to the left of $i$ .

$x = \frac{- 4 \pm 2 i \sqrt{11}}{2 \cdot 3}$

Step 1.2.2.3.2

Multiply $2$ by $3$ .

$x = \frac{- 4 \pm 2 i \sqrt{11}}{6}$

Step 1.2.2.3.3

Simplify $\frac{- 4 \pm 2 i \sqrt{11}}{6}$ .

$x = \frac{- 2 \pm i \sqrt{11}}{3}$

Step 1.2.2.4

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

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Step 1.2.2.4.1

Simplify the numerator.

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Step 1.2.2.4.1.1

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 3 \cdot 5}}{2 \cdot 3}$

Step 1.2.2.4.1.2

Multiply $- 4 \cdot 3 \cdot 5$ .

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Step 1.2.2.4.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{- 4 \pm \sqrt{16 - 12 \cdot 5}}{2 \cdot 3}$

Step 1.2.2.4.1.2.2

Multiply $- 12$ by $5$ .

$x = \frac{- 4 \pm \sqrt{16 - 60}}{2 \cdot 3}$

Step 1.2.2.4.1.3

Subtract $60$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 44}}{2 \cdot 3}$

Step 1.2.2.4.1.4

Rewrite $- 44$ as $- 1 (44)$ .

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 44}}{2 \cdot 3}$

Step 1.2.2.4.1.5

Rewrite $\sqrt{- 1 (44)}$ as $\sqrt{- 1} \cdot \sqrt{44}$ .

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{44}}{2 \cdot 3}$

Step 1.2.2.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 4 \pm i \cdot \sqrt{44}}{2 \cdot 3}$

Step 1.2.2.4.1.7

Rewrite $44$ as $2^{2} \cdot 11$ .

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Step 1.2.2.4.1.7.1

Factor $4$ out of $44$ .

$x = \frac{- 4 \pm i \cdot \sqrt{4 (11)}}{2 \cdot 3}$

Step 1.2.2.4.1.7.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 4 \pm i \cdot \sqrt{2^{2} \cdot 11}}{2 \cdot 3}$

Step 1.2.2.4.1.8

Pull terms out from under the radical.

$x = \frac{- 4 \pm i \cdot (2 \sqrt{11})}{2 \cdot 3}$

Step 1.2.2.4.1.9

Move $2$ to the left of $i$ .

$x = \frac{- 4 \pm 2 i \sqrt{11}}{2 \cdot 3}$

Step 1.2.2.4.2

Multiply $2$ by $3$ .

$x = \frac{- 4 \pm 2 i \sqrt{11}}{6}$

Step 1.2.2.4.3

Simplify $\frac{- 4 \pm 2 i \sqrt{11}}{6}$ .

$x = \frac{- 2 \pm i \sqrt{11}}{3}$

Step 1.2.2.4.4

Change the $\pm$ to $+$ .

$x = \frac{- 2 + i \sqrt{11}}{3}$

Step 1.2.2.4.5

Rewrite $- 2$ as $- 1 (2)$ .

$x = \frac{- 1 \cdot 2 + i \sqrt{11}}{3}$

Step 1.2.2.4.6

Factor $- 1$ out of $i \sqrt{11}$ .

$x = \frac{- 1 \cdot 2 - (- i \sqrt{11})}{3}$

Step 1.2.2.4.7

Factor $- 1$ out of $- 1 (2) - (- i \sqrt{11})$ .

$x = \frac{- 1 (2 - i \sqrt{11})}{3}$

Step 1.2.2.4.8

Move the negative in front of the fraction.

$x = - \frac{2 - i \sqrt{11}}{3}$

Step 1.2.2.5

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

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Step 1.2.2.5.1

Simplify the numerator.

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Step 1.2.2.5.1.1

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 3 \cdot 5}}{2 \cdot 3}$

Step 1.2.2.5.1.2

Multiply $- 4 \cdot 3 \cdot 5$ .

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Step 1.2.2.5.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{- 4 \pm \sqrt{16 - 12 \cdot 5}}{2 \cdot 3}$

Step 1.2.2.5.1.2.2

Multiply $- 12$ by $5$ .

$x = \frac{- 4 \pm \sqrt{16 - 60}}{2 \cdot 3}$

Step 1.2.2.5.1.3

Subtract $60$ from $16$ .

$x = \frac{- 4 \pm \sqrt{- 44}}{2 \cdot 3}$

Step 1.2.2.5.1.4

Rewrite $- 44$ as $- 1 (44)$ .

$x = \frac{- 4 \pm \sqrt{- 1 \cdot 44}}{2 \cdot 3}$

Step 1.2.2.5.1.5

Rewrite $\sqrt{- 1 (44)}$ as $\sqrt{- 1} \cdot \sqrt{44}$ .

$x = \frac{- 4 \pm \sqrt{- 1} \cdot \sqrt{44}}{2 \cdot 3}$

Step 1.2.2.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 4 \pm i \cdot \sqrt{44}}{2 \cdot 3}$

Step 1.2.2.5.1.7

Rewrite $44$ as $2^{2} \cdot 11$ .

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Step 1.2.2.5.1.7.1

Factor $4$ out of $44$ .

$x = \frac{- 4 \pm i \cdot \sqrt{4 (11)}}{2 \cdot 3}$

Step 1.2.2.5.1.7.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 4 \pm i \cdot \sqrt{2^{2} \cdot 11}}{2 \cdot 3}$

Step 1.2.2.5.1.8

Pull terms out from under the radical.

$x = \frac{- 4 \pm i \cdot (2 \sqrt{11})}{2 \cdot 3}$

Step 1.2.2.5.1.9

Move $2$ to the left of $i$ .

$x = \frac{- 4 \pm 2 i \sqrt{11}}{2 \cdot 3}$

Step 1.2.2.5.2

Multiply $2$ by $3$ .

$x = \frac{- 4 \pm 2 i \sqrt{11}}{6}$

Step 1.2.2.5.3

Simplify $\frac{- 4 \pm 2 i \sqrt{11}}{6}$ .

$x = \frac{- 2 \pm i \sqrt{11}}{3}$

Step 1.2.2.5.4

Change the $\pm$ to $-$ .

$x = \frac{- 2 - i \sqrt{11}}{3}$

Step 1.2.2.5.5

Rewrite $- 2$ as $- 1 (2)$ .

$x = \frac{- 1 \cdot 2 - i \sqrt{11}}{3}$

Step 1.2.2.5.6

Factor $- 1$ out of $- i \sqrt{11}$ .

$x = \frac{- 1 \cdot 2 - (i \sqrt{11})}{3}$

Step 1.2.2.5.7

Factor $- 1$ out of $- 1 (2) - (i \sqrt{11})$ .

$x = \frac{- 1 (2 + i \sqrt{11})}{3}$

Step 1.2.2.5.8

Move the negative in front of the fraction.

$x = - \frac{2 + i \sqrt{11}}{3}$

Step 1.2.2.6

The final answer is the combination of both solutions.

$x = - \frac{2 - i \sqrt{11}}{3}, - \frac{2 + i \sqrt{11}}{3}$

Step 1.3

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

x-intercept(s): $None$

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

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

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

Remove parentheses.

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

Step 2.2.4

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

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Step 2.2.4.1

Simplify the numerator.

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Step 2.2.4.1.1

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

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

Step 2.2.4.1.2

Multiply $3$ by $0$ .

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

Step 2.2.4.1.3

Multiply $4$ by $0$ .

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

Step 2.2.4.1.4

Add $0$ and $0$ .

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

Step 2.2.4.1.5

Add $0$ and $5$ .

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

Step 2.2.4.2

Simplify the denominator.

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Step 2.2.4.2.1

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

$y = \frac{5}{5 \cdot 0 + 2 (0) - 3}$

Step 2.2.4.2.2

Multiply $5$ by $0$ .

$y = \frac{5}{0 + 2 (0) - 3}$

Step 2.2.4.2.3

Multiply $2$ by $0$ .

$y = \frac{5}{0 + 0 - 3}$

Step 2.2.4.2.4

Add $0$ and $0$ .

$y = \frac{5}{0 - 3}$

Step 2.2.4.2.5

Subtract $3$ from $0$ .

$y = \frac{5}{- 3}$

Step 2.2.4.3

Move the negative in front of the fraction.

$y = - \frac{5}{3}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{5}{3})$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, - \frac{5}{3})$

Step 4