Find the x and y Intercepts 4x^2-24x+31

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

Step 1

Write $4 x^{2} - 24 x + 31$ as an equation.

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

Step 2

Find the x-intercepts.

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

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

Solve the equation.

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

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

Use the quadratic formula to find the solutions.

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

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

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

Simplify.

Simplify the numerator.

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

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

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

Multiply $- 4$ by $4$ .

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

Multiply $- 16$ by $31$ .

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

Subtract $496$ from $576$ .

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

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

Factor $16$ out of $80$ .

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $4$ .

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

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

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

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

Simplify the numerator.

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

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

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

Multiply $- 4$ by $4$ .

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

Multiply $- 16$ by $31$ .

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

Subtract $496$ from $576$ .

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

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

Factor $16$ out of $80$ .

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $4$ .

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

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

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

Change the $\pm$ to $+$ .

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

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

Simplify the numerator.

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

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

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

Multiply $- 4$ by $4$ .

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

Multiply $- 16$ by $31$ .

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

Subtract $496$ from $576$ .

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

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

Factor $16$ out of $80$ .

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $4$ .

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

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

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

Change the $\pm$ to $-$ .

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

The final answer is the combination of both solutions.

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

x-intercept(s) in point form.

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

Step 3

Find the y-intercepts.

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

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

Solve the equation.

Remove parentheses.

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

Remove parentheses.

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

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

Simplify each term.

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

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

Multiply $4$ by $0$ .

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

Multiply $- 24$ by $0$ .

$y = 0 + 0 + 31$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + 31$

Add $0$ and $31$ .

$y = 31$

y-intercept(s) in point form.

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

Step 4

List the intersections.

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

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

Step 5

Find the x and y Intercepts 4x^2-24x+31

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