Find the x and y Intercepts y=-x^2+4x+8

$y = - x^{2} + 4 x + 8$

Step 1

Find the x-intercepts.

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

$0 = - x^{2} + 4 x + 8$

Solve the equation.

Rewrite the equation as $- x^{2} + 4 x + 8 = 0$ .

$- x^{2} + 4 x + 8 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = - 1$ , $b = 4$ , and $c = 8$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (- 1 \cdot 8)}}{2 \cdot - 1}$

Simplify.

Simplify the numerator.

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

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 1 \cdot 8}}{2 \cdot - 1}$

Multiply $- 4 \cdot - 1 \cdot 8$ .

Multiply $- 4$ by $- 1$ .

$x = \frac{- 4 \pm \sqrt{16 + 4 \cdot 8}}{2 \cdot - 1}$

Multiply $4$ by $8$ .

$x = \frac{- 4 \pm \sqrt{16 + 32}}{2 \cdot - 1}$

Add $16$ and $32$ .

$x = \frac{- 4 \pm \sqrt{48}}{2 \cdot - 1}$

Rewrite $48$ as $4^{2} \cdot 3$ .

Factor $16$ out of $48$ .

$x = \frac{- 4 \pm \sqrt{16 (3)}}{2 \cdot - 1}$

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $- 1$ .

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

Simplify $\frac{- 4 \pm 4 \sqrt{3}}{- 2}$ .

$x = 2 \pm 2 \sqrt{3}$

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

Simplify the numerator.

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

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 1 \cdot 8}}{2 \cdot - 1}$

Multiply $- 4 \cdot - 1 \cdot 8$ .

Multiply $- 4$ by $- 1$ .

$x = \frac{- 4 \pm \sqrt{16 + 4 \cdot 8}}{2 \cdot - 1}$

Multiply $4$ by $8$ .

$x = \frac{- 4 \pm \sqrt{16 + 32}}{2 \cdot - 1}$

Add $16$ and $32$ .

$x = \frac{- 4 \pm \sqrt{48}}{2 \cdot - 1}$

Rewrite $48$ as $4^{2} \cdot 3$ .

Factor $16$ out of $48$ .

$x = \frac{- 4 \pm \sqrt{16 (3)}}{2 \cdot - 1}$

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $- 1$ .

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

Simplify $\frac{- 4 \pm 4 \sqrt{3}}{- 2}$ .

$x = 2 \pm 2 \sqrt{3}$

Change the $\pm$ to $+$ .

$x = 2 + 2 \sqrt{3}$

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

Simplify the numerator.

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

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 1 \cdot 8}}{2 \cdot - 1}$

Multiply $- 4 \cdot - 1 \cdot 8$ .

Multiply $- 4$ by $- 1$ .

$x = \frac{- 4 \pm \sqrt{16 + 4 \cdot 8}}{2 \cdot - 1}$

Multiply $4$ by $8$ .

$x = \frac{- 4 \pm \sqrt{16 + 32}}{2 \cdot - 1}$

Add $16$ and $32$ .

$x = \frac{- 4 \pm \sqrt{48}}{2 \cdot - 1}$

Rewrite $48$ as $4^{2} \cdot 3$ .

Factor $16$ out of $48$ .

$x = \frac{- 4 \pm \sqrt{16 (3)}}{2 \cdot - 1}$

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $- 1$ .

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

Simplify $\frac{- 4 \pm 4 \sqrt{3}}{- 2}$ .

$x = 2 \pm 2 \sqrt{3}$

Change the $\pm$ to $-$ .

$x = 2 - 2 \sqrt{3}$

The final answer is the combination of both solutions.

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

x-intercept(s) in point form.

x-intercept(s): $(2 + 2 \sqrt{3}, 0), (2 - 2 \sqrt{3}, 0)$

Step 2

Find the y-intercepts.

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

$y = - {(0)}^{2} + 4 (0) + 8$

Solve the equation.

Remove parentheses.

$y = - 0^{2} + 4 (0) + 8$

Remove parentheses.

$y = - {(0)}^{2} + 4 (0) + 8$

Simplify $- {(0)}^{2} + 4 (0) + 8$ .

Simplify each term.

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

$y = - 0 + 4 (0) + 8$

Multiply $- 1$ by $0$ .

$y = 0 + 4 (0) + 8$

Multiply $4$ by $0$ .

$y = 0 + 0 + 8$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + 8$

Add $0$ and $8$ .

$y = 8$

y-intercept(s) in point form.

y-intercept(s): $(0, 8)$

Step 3

List the intersections.

x-intercept(s): $(2 + 2 \sqrt{3}, 0), (2 - 2 \sqrt{3}, 0)$

y-intercept(s): $(0, 8)$

Step 4

Find the x and y Intercepts y=-x^2+4x+8

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