Find the x and y Intercepts -x^2+6x-18

$- x^{2} + 6 x - 18$

Step 1

Write $- x^{2} + 6 x - 18$ as an equation.

$y = - x^{2} + 6 x - 18$

Step 2

Find the x-intercepts.

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

$0 = - x^{2} + 6 x - 18$

Solve the equation.

Rewrite the equation as $- x^{2} + 6 x - 18 = 0$ .

$- x^{2} + 6 x - 18 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = - 1$ , $b = 6$ , and $c = - 18$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (- 1 \cdot - 18)}}{2 \cdot - 1}$

Simplify.

Simplify the numerator.

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 1 \cdot - 18}}{2 \cdot - 1}$

Multiply $- 4 \cdot - 1 \cdot - 18$ .

Multiply $- 4$ by $- 1$ .

$x = \frac{- 6 \pm \sqrt{36 + 4 \cdot - 18}}{2 \cdot - 1}$

Multiply $4$ by $- 18$ .

$x = \frac{- 6 \pm \sqrt{36 - 72}}{2 \cdot - 1}$

Subtract $72$ from $36$ .

$x = \frac{- 6 \pm \sqrt{- 36}}{2 \cdot - 1}$

Rewrite $- 36$ as $- 1 (36)$ .

$x = \frac{- 6 \pm \sqrt{- 1 \cdot 36}}{2 \cdot - 1}$

Rewrite $\sqrt{- 1 (36)}$ as $\sqrt{- 1} \cdot \sqrt{36}$ .

$x = \frac{- 6 \pm \sqrt{- 1} \cdot \sqrt{36}}{2 \cdot - 1}$

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

$x = \frac{- 6 \pm i \cdot \sqrt{36}}{2 \cdot - 1}$

Rewrite $36$ as $6^{2}$ .

$x = \frac{- 6 \pm i \cdot \sqrt{6^{2}}}{2 \cdot - 1}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 6 \pm i \cdot 6}{2 \cdot - 1}$

Move $6$ to the left of $i$ .

$x = \frac{- 6 \pm 6 i}{2 \cdot - 1}$

Multiply $2$ by $- 1$ .

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

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

$x = 3 \pm 3 i$

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

Simplify the numerator.

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 1 \cdot - 18}}{2 \cdot - 1}$

Multiply $- 4 \cdot - 1 \cdot - 18$ .

Multiply $- 4$ by $- 1$ .

$x = \frac{- 6 \pm \sqrt{36 + 4 \cdot - 18}}{2 \cdot - 1}$

Multiply $4$ by $- 18$ .

$x = \frac{- 6 \pm \sqrt{36 - 72}}{2 \cdot - 1}$

Subtract $72$ from $36$ .

$x = \frac{- 6 \pm \sqrt{- 36}}{2 \cdot - 1}$

Rewrite $- 36$ as $- 1 (36)$ .

$x = \frac{- 6 \pm \sqrt{- 1 \cdot 36}}{2 \cdot - 1}$

Rewrite $\sqrt{- 1 (36)}$ as $\sqrt{- 1} \cdot \sqrt{36}$ .

$x = \frac{- 6 \pm \sqrt{- 1} \cdot \sqrt{36}}{2 \cdot - 1}$

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

$x = \frac{- 6 \pm i \cdot \sqrt{36}}{2 \cdot - 1}$

Rewrite $36$ as $6^{2}$ .

$x = \frac{- 6 \pm i \cdot \sqrt{6^{2}}}{2 \cdot - 1}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 6 \pm i \cdot 6}{2 \cdot - 1}$

Move $6$ to the left of $i$ .

$x = \frac{- 6 \pm 6 i}{2 \cdot - 1}$

Multiply $2$ by $- 1$ .

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

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

$x = 3 \pm 3 i$

Change the $\pm$ to $+$ .

$x = 3 + 3 i$

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

Simplify the numerator.

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 1 \cdot - 18}}{2 \cdot - 1}$

Multiply $- 4 \cdot - 1 \cdot - 18$ .

Multiply $- 4$ by $- 1$ .

$x = \frac{- 6 \pm \sqrt{36 + 4 \cdot - 18}}{2 \cdot - 1}$

Multiply $4$ by $- 18$ .

$x = \frac{- 6 \pm \sqrt{36 - 72}}{2 \cdot - 1}$

Subtract $72$ from $36$ .

$x = \frac{- 6 \pm \sqrt{- 36}}{2 \cdot - 1}$

Rewrite $- 36$ as $- 1 (36)$ .

$x = \frac{- 6 \pm \sqrt{- 1 \cdot 36}}{2 \cdot - 1}$

Rewrite $\sqrt{- 1 (36)}$ as $\sqrt{- 1} \cdot \sqrt{36}$ .

$x = \frac{- 6 \pm \sqrt{- 1} \cdot \sqrt{36}}{2 \cdot - 1}$

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

$x = \frac{- 6 \pm i \cdot \sqrt{36}}{2 \cdot - 1}$

Rewrite $36$ as $6^{2}$ .

$x = \frac{- 6 \pm i \cdot \sqrt{6^{2}}}{2 \cdot - 1}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 6 \pm i \cdot 6}{2 \cdot - 1}$

Move $6$ to the left of $i$ .

$x = \frac{- 6 \pm 6 i}{2 \cdot - 1}$

Multiply $2$ by $- 1$ .

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

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

$x = 3 \pm 3 i$

Change the $\pm$ to $-$ .

$x = 3 - 3 i$

The final answer is the combination of both solutions.

$x = 3 + 3 i, 3 - 3 i$

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

x-intercept(s): $None$

Step 3

Find the y-intercepts.

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

$y = - {(0)}^{2} + 6 (0) - 18$

Solve the equation.

Remove parentheses.

$y = - 0^{2} + 6 (0) - 18$

Remove parentheses.

$y = - {(0)}^{2} + 6 (0) - 18$

Simplify $- {(0)}^{2} + 6 (0) - 18$ .

Simplify each term.

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

$y = - 0 + 6 (0) - 18$

Multiply $- 1$ by $0$ .

$y = 0 + 6 (0) - 18$

Multiply $6$ by $0$ .

$y = 0 + 0 - 18$

Simplify by adding and subtracting.

Add $0$ and $0$ .

$y = 0 - 18$

Subtract $18$ from $0$ .

$y = - 18$

y-intercept(s) in point form.

y-intercept(s): $(0, - 18)$

Step 4

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, - 18)$

Step 5

Find the x and y Intercepts -x^2+6x-18

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