Calculus Examples

$y = 4 x^{2} - 24 x + 31$

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 $x$ .

$0 = 4 x^{2} - 24 x + 31$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $4 x^{2} - 24 x + 31 = 0$ .

$4 x^{2} - 24 x + 31 = 0$

Step 1.2.2

Use the quadratic formula to find the solutions.

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

Step 1.2.3

Substitute the values $a = 4$ , $b = - 24$ , and $c = 31$ into the quadratic formula and solve for $x$ .

$\frac{24 \pm \sqrt{{(- 24)}^{2} - 4 \cdot (4 \cdot 31)}}{2 \cdot 4}$

Step 1.2.4

Simplify.

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Step 1.2.4.1

Simplify the numerator.

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Step 1.2.4.1.1

Raise $- 24$ to the power of $2$ .

$x = \frac{24 \pm \sqrt{576 - 4 \cdot 4 \cdot 31}}{2 \cdot 4}$

Step 1.2.4.1.2

Multiply $- 4 \cdot 4 \cdot 31$ .

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Step 1.2.4.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{24 \pm \sqrt{576 - 16 \cdot 31}}{2 \cdot 4}$

Step 1.2.4.1.2.2

Multiply $- 16$ by $31$ .

$x = \frac{24 \pm \sqrt{576 - 496}}{2 \cdot 4}$

Step 1.2.4.1.3

Subtract $496$ from $576$ .

$x = \frac{24 \pm \sqrt{80}}{2 \cdot 4}$

Step 1.2.4.1.4

Rewrite $80$ as $4^{2} \cdot 5$ .

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Step 1.2.4.1.4.1

Factor $16$ out of $80$ .

$x = \frac{24 \pm \sqrt{16 (5)}}{2 \cdot 4}$

Step 1.2.4.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{24 \pm \sqrt{4^{2} \cdot 5}}{2 \cdot 4}$

Step 1.2.4.1.5

Pull terms out from under the radical.

$x = \frac{24 \pm 4 \sqrt{5}}{2 \cdot 4}$

Step 1.2.4.2

Multiply $2$ by $4$ .

$x = \frac{24 \pm 4 \sqrt{5}}{8}$

Step 1.2.4.3

Simplify $\frac{24 \pm 4 \sqrt{5}}{8}$ .

$x = \frac{6 \pm \sqrt{5}}{2}$

Step 1.2.5

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

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Step 1.2.5.1

Simplify the numerator.

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Step 1.2.5.1.1

Raise $- 24$ to the power of $2$ .

$x = \frac{24 \pm \sqrt{576 - 4 \cdot 4 \cdot 31}}{2 \cdot 4}$

Step 1.2.5.1.2

Multiply $- 4 \cdot 4 \cdot 31$ .

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Step 1.2.5.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{24 \pm \sqrt{576 - 16 \cdot 31}}{2 \cdot 4}$

Step 1.2.5.1.2.2

Multiply $- 16$ by $31$ .

$x = \frac{24 \pm \sqrt{576 - 496}}{2 \cdot 4}$

Step 1.2.5.1.3

Subtract $496$ from $576$ .

$x = \frac{24 \pm \sqrt{80}}{2 \cdot 4}$

Step 1.2.5.1.4

Rewrite $80$ as $4^{2} \cdot 5$ .

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Step 1.2.5.1.4.1

Factor $16$ out of $80$ .

$x = \frac{24 \pm \sqrt{16 (5)}}{2 \cdot 4}$

Step 1.2.5.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{24 \pm \sqrt{4^{2} \cdot 5}}{2 \cdot 4}$

Step 1.2.5.1.5

Pull terms out from under the radical.

$x = \frac{24 \pm 4 \sqrt{5}}{2 \cdot 4}$

Step 1.2.5.2

Multiply $2$ by $4$ .

$x = \frac{24 \pm 4 \sqrt{5}}{8}$

Step 1.2.5.3

Simplify $\frac{24 \pm 4 \sqrt{5}}{8}$ .

$x = \frac{6 \pm \sqrt{5}}{2}$

Step 1.2.5.4

Change the $\pm$ to $+$ .

$x = \frac{6 + \sqrt{5}}{2}$

Step 1.2.6

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

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Step 1.2.6.1

Simplify the numerator.

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Step 1.2.6.1.1

Raise $- 24$ to the power of $2$ .

$x = \frac{24 \pm \sqrt{576 - 4 \cdot 4 \cdot 31}}{2 \cdot 4}$

Step 1.2.6.1.2

Multiply $- 4 \cdot 4 \cdot 31$ .

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Step 1.2.6.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{24 \pm \sqrt{576 - 16 \cdot 31}}{2 \cdot 4}$

Step 1.2.6.1.2.2

Multiply $- 16$ by $31$ .

$x = \frac{24 \pm \sqrt{576 - 496}}{2 \cdot 4}$

Step 1.2.6.1.3

Subtract $496$ from $576$ .

$x = \frac{24 \pm \sqrt{80}}{2 \cdot 4}$

Step 1.2.6.1.4

Rewrite $80$ as $4^{2} \cdot 5$ .

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Step 1.2.6.1.4.1

Factor $16$ out of $80$ .

$x = \frac{24 \pm \sqrt{16 (5)}}{2 \cdot 4}$

Step 1.2.6.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{24 \pm \sqrt{4^{2} \cdot 5}}{2 \cdot 4}$

Step 1.2.6.1.5

Pull terms out from under the radical.

$x = \frac{24 \pm 4 \sqrt{5}}{2 \cdot 4}$

Step 1.2.6.2

Multiply $2$ by $4$ .

$x = \frac{24 \pm 4 \sqrt{5}}{8}$

Step 1.2.6.3

Simplify $\frac{24 \pm 4 \sqrt{5}}{8}$ .

$x = \frac{6 \pm \sqrt{5}}{2}$

Step 1.2.6.4

Change the $\pm$ to $-$ .

$x = \frac{6 - \sqrt{5}}{2}$

Step 1.2.7

The final answer is the combination of both solutions.

$x = \frac{6 + \sqrt{5}}{2}, \frac{6 - \sqrt{5}}{2}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{6 + \sqrt{5}}{2}, 0), (\frac{6 - \sqrt{5}}{2}, 0)$

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 $y$ .

$y = 4 {(0)}^{2} - 24 \cdot 0 + 31$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 4 \cdot 0^{2} - 24 \cdot 0 + 31$

Step 2.2.2

Remove parentheses.

$y = 4 {(0)}^{2} - 24 \cdot 0 + 31$

Step 2.2.3

Simplify $4 {(0)}^{2} - 24 \cdot 0 + 31$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

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

$y = 4 \cdot 0 - 24 \cdot 0 + 31$

Step 2.2.3.1.2

Multiply $4$ by $0$ .

$y = 0 - 24 \cdot 0 + 31$

Step 2.2.3.1.3

Multiply $- 24$ by $0$ .

$y = 0 + 0 + 31$

Step 2.2.3.2

Simplify by adding numbers.

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Step 2.2.3.2.1

Add $0$ and $0$ .

$y = 0 + 31$

Step 2.2.3.2.2

Add $0$ and $31$ .

$y = 31$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 31)$

Step 3

List the intersections.

x-intercept(s): $(\frac{6 + \sqrt{5}}{2}, 0), (\frac{6 - \sqrt{5}}{2}, 0)$

y-intercept(s): $(0, 31)$

Step 4