Find the x and y Intercepts -x^2-2x-5

$- x^{2} - 2 x - 5$

Step 1

Write $- x^{2} - 2 x - 5$ as an equation.

$y = - x^{2} - 2 x - 5$

Step 2

Find the x-intercepts.

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

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

Solve the equation.

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

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

Use the quadratic formula to find the solutions.

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

Substitute the values $a = - 1$ , $b = - 2$ , and $c = - 5$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (- 1 \cdot - 5)}}{2 \cdot - 1}$

Simplify.

Simplify the numerator.

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot - 1 \cdot - 5}}{2 \cdot - 1}$

Multiply $- 4 \cdot - 1 \cdot - 5$ .

Multiply $- 4$ by $- 1$ .

$x = \frac{2 \pm \sqrt{4 + 4 \cdot - 5}}{2 \cdot - 1}$

Multiply $4$ by $- 5$ .

$x = \frac{2 \pm \sqrt{4 - 20}}{2 \cdot - 1}$

Subtract $20$ from $4$ .

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

Rewrite $- 16$ as $- 1 (16)$ .

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

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

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

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

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

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

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

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

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

Move $4$ to the left of $i$ .

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

Multiply $2$ by $- 1$ .

$x = \frac{2 \pm 4 i}{- 2}$

Simplify $\frac{2 \pm 4 i}{- 2}$ .

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

Move the negative one from the denominator of $\frac{1 \pm 2 i}{- 1}$ .

$x = - 1 \cdot (1 \pm 2 i)$

Rewrite $- 1 \cdot (1 \pm 2 i)$ as $- (1 \pm 2 i)$ .

$x = - (1 \pm 2 i)$

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

Simplify the numerator.

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot - 1 \cdot - 5}}{2 \cdot - 1}$

Multiply $- 4 \cdot - 1 \cdot - 5$ .

Multiply $- 4$ by $- 1$ .

$x = \frac{2 \pm \sqrt{4 + 4 \cdot - 5}}{2 \cdot - 1}$

Multiply $4$ by $- 5$ .

$x = \frac{2 \pm \sqrt{4 - 20}}{2 \cdot - 1}$

Subtract $20$ from $4$ .

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

Rewrite $- 16$ as $- 1 (16)$ .

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

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

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

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

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

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

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

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

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

Move $4$ to the left of $i$ .

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

Multiply $2$ by $- 1$ .

$x = \frac{2 \pm 4 i}{- 2}$

Simplify $\frac{2 \pm 4 i}{- 2}$ .

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

Move the negative one from the denominator of $\frac{1 \pm 2 i}{- 1}$ .

$x = - 1 \cdot (1 \pm 2 i)$

Rewrite $- 1 \cdot (1 \pm 2 i)$ as $- (1 \pm 2 i)$ .

$x = - (1 \pm 2 i)$

Change the $\pm$ to $+$ .

$x = - (1 + 2 i)$

Apply the distributive property.

$x = - 1 \cdot 1 - (2 i)$

Multiply $- 1$ by $1$ .

$x = - 1 - (2 i)$

Multiply $2$ by $- 1$ .

$x = - 1 - 2 i$

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

Simplify the numerator.

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot - 1 \cdot - 5}}{2 \cdot - 1}$

Multiply $- 4 \cdot - 1 \cdot - 5$ .

Multiply $- 4$ by $- 1$ .

$x = \frac{2 \pm \sqrt{4 + 4 \cdot - 5}}{2 \cdot - 1}$

Multiply $4$ by $- 5$ .

$x = \frac{2 \pm \sqrt{4 - 20}}{2 \cdot - 1}$

Subtract $20$ from $4$ .

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

Rewrite $- 16$ as $- 1 (16)$ .

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

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

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

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

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

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

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

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

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

Move $4$ to the left of $i$ .

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

Multiply $2$ by $- 1$ .

$x = \frac{2 \pm 4 i}{- 2}$

Simplify $\frac{2 \pm 4 i}{- 2}$ .

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

Move the negative one from the denominator of $\frac{1 \pm 2 i}{- 1}$ .

$x = - 1 \cdot (1 \pm 2 i)$

Rewrite $- 1 \cdot (1 \pm 2 i)$ as $- (1 \pm 2 i)$ .

$x = - (1 \pm 2 i)$

Change the $\pm$ to $-$ .

$x = - (1 - 2 i)$

Apply the distributive property.

$x = - 1 \cdot 1 - (- 2 i)$

Multiply $- 1$ by $1$ .

$x = - 1 - (- 2 i)$

Multiply $- 2$ by $- 1$ .

$x = - 1 + 2 i$

The final answer is the combination of both solutions.

$x = - 1 - 2 i, - 1 + 2 i$

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 $y$ .

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

Solve the equation.

Remove parentheses.

$y = - 0^{2} - 2 \cdot 0 - 5$

Remove parentheses.

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

Simplify $- {(0)}^{2} - 2 \cdot 0 - 5$ .

Simplify each term.

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

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

Multiply $- 1$ by $0$ .

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

Multiply $- 2$ by $0$ .

$y = 0 + 0 - 5$

Simplify by adding and subtracting.

Add $0$ and $0$ .

$y = 0 - 5$

Subtract $5$ from $0$ .

$y = - 5$

y-intercept(s) in point form.

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

Step 4

List the intersections.

x-intercept(s): $None$

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

Step 5

Find the x and y Intercepts -x^2-2x-5

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