Find the x and y Intercepts f(x)=-5+8x-2x^2

$f (x) = - 5 + 8 x - 2 x^{2}$

Step 1

Find the x-intercepts.

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

$0 = - 5 + 8 x - 2 x^{2}$

Solve the equation.

Rewrite the equation as $- 5 + 8 x - 2 x^{2} = 0$ .

$- 5 + 8 x - 2 x^{2} = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = - 2$ , $b = 8$ , and $c = - 5$ into the quadratic formula and solve for $x$ .

$\frac{- 8 \pm \sqrt{8^{2} - 4 \cdot (- 2 \cdot - 5)}}{2 \cdot - 2}$

Simplify.

Simplify the numerator.

Raise $8$ to the power of $2$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot - 2 \cdot - 5}}{2 \cdot - 2}$

Multiply $- 4 \cdot - 2 \cdot - 5$ .

Multiply $- 4$ by $- 2$ .

$x = \frac{- 8 \pm \sqrt{64 + 8 \cdot - 5}}{2 \cdot - 2}$

Multiply $8$ by $- 5$ .

$x = \frac{- 8 \pm \sqrt{64 - 40}}{2 \cdot - 2}$

Subtract $40$ from $64$ .

$x = \frac{- 8 \pm \sqrt{24}}{2 \cdot - 2}$

Rewrite $24$ as $2^{2} \cdot 6$ .

Factor $4$ out of $24$ .

$x = \frac{- 8 \pm \sqrt{4 (6)}}{2 \cdot - 2}$

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

$x = \frac{- 8 \pm \sqrt{2^{2} \cdot 6}}{2 \cdot - 2}$

Pull terms out from under the radical.

$x = \frac{- 8 \pm 2 \sqrt{6}}{2 \cdot - 2}$

Multiply $2$ by $- 2$ .

$x = \frac{- 8 \pm 2 \sqrt{6}}{- 4}$

Simplify $\frac{- 8 \pm 2 \sqrt{6}}{- 4}$ .

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

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

Simplify the numerator.

Raise $8$ to the power of $2$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot - 2 \cdot - 5}}{2 \cdot - 2}$

Multiply $- 4 \cdot - 2 \cdot - 5$ .

Multiply $- 4$ by $- 2$ .

$x = \frac{- 8 \pm \sqrt{64 + 8 \cdot - 5}}{2 \cdot - 2}$

Multiply $8$ by $- 5$ .

$x = \frac{- 8 \pm \sqrt{64 - 40}}{2 \cdot - 2}$

Subtract $40$ from $64$ .

$x = \frac{- 8 \pm \sqrt{24}}{2 \cdot - 2}$

Rewrite $24$ as $2^{2} \cdot 6$ .

Factor $4$ out of $24$ .

$x = \frac{- 8 \pm \sqrt{4 (6)}}{2 \cdot - 2}$

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

$x = \frac{- 8 \pm \sqrt{2^{2} \cdot 6}}{2 \cdot - 2}$

Pull terms out from under the radical.

$x = \frac{- 8 \pm 2 \sqrt{6}}{2 \cdot - 2}$

Multiply $2$ by $- 2$ .

$x = \frac{- 8 \pm 2 \sqrt{6}}{- 4}$

Simplify $\frac{- 8 \pm 2 \sqrt{6}}{- 4}$ .

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

Change the $\pm$ to $+$ .

$x = \frac{4 + \sqrt{6}}{2}$

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

Simplify the numerator.

Raise $8$ to the power of $2$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot - 2 \cdot - 5}}{2 \cdot - 2}$

Multiply $- 4 \cdot - 2 \cdot - 5$ .

Multiply $- 4$ by $- 2$ .

$x = \frac{- 8 \pm \sqrt{64 + 8 \cdot - 5}}{2 \cdot - 2}$

Multiply $8$ by $- 5$ .

$x = \frac{- 8 \pm \sqrt{64 - 40}}{2 \cdot - 2}$

Subtract $40$ from $64$ .

$x = \frac{- 8 \pm \sqrt{24}}{2 \cdot - 2}$

Rewrite $24$ as $2^{2} \cdot 6$ .

Factor $4$ out of $24$ .

$x = \frac{- 8 \pm \sqrt{4 (6)}}{2 \cdot - 2}$

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

$x = \frac{- 8 \pm \sqrt{2^{2} \cdot 6}}{2 \cdot - 2}$

Pull terms out from under the radical.

$x = \frac{- 8 \pm 2 \sqrt{6}}{2 \cdot - 2}$

Multiply $2$ by $- 2$ .

$x = \frac{- 8 \pm 2 \sqrt{6}}{- 4}$

Simplify $\frac{- 8 \pm 2 \sqrt{6}}{- 4}$ .

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

Change the $\pm$ to $-$ .

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

The final answer is the combination of both solutions.

$x = \frac{4 + \sqrt{6}}{2}, \frac{4 - \sqrt{6}}{2}$

x-intercept(s) in point form.

x-intercept(s): $(\frac{4 + \sqrt{6}}{2}, 0), (\frac{4 - \sqrt{6}}{2}, 0)$

Step 2

Find the y-intercepts.

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

$y = - 5 + 8 (0) - 2 {(0)}^{2}$

Solve the equation.

Remove parentheses.

$y = - 5 + 8 (0) - 2 \cdot 0^{2}$

Remove parentheses.

$y = - 5 + 8 (0) - 2 {(0)}^{2}$

Simplify $- 5 + 8 (0) - 2 {(0)}^{2}$ .

Simplify each term.

Multiply $8$ by $0$ .

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

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

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

Multiply $- 2$ by $0$ .

$y = - 5 + 0 + 0$

Simplify by adding numbers.

Add $- 5$ and $0$ .

$y = - 5 + 0$

Add $- 5$ and $0$ .

$y = - 5$

y-intercept(s) in point form.

y-intercept(s): $(0, - 5)$

Step 3

List the intersections.

x-intercept(s): $(\frac{4 + \sqrt{6}}{2}, 0), (\frac{4 - \sqrt{6}}{2}, 0)$

y-intercept(s): $(0, - 5)$

Step 4

Find the x and y Intercepts f(x)=-5+8x-2x^2

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