Precalculus Examples

$x = - y^{2} - 8 y - 22$

Step 1

Find the x-intercepts.

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Step 1.1

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

$x = - {(0)}^{2} - 8 \cdot 0 - 22$

Step 1.2

Solve the equation.

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Step 1.2.1

Remove parentheses.

$x = - 0^{2} - 8 \cdot 0 - 22$

Step 1.2.2

Remove parentheses.

$x = - {(0)}^{2} - 8 \cdot 0 - 22$

Step 1.2.3

Simplify $- {(0)}^{2} - 8 \cdot 0 - 22$ .

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

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

$x = - 0 - 8 \cdot 0 - 22$

Step 1.2.3.1.2

Multiply $- 1$ by $0$ .

$x = 0 - 8 \cdot 0 - 22$

Step 1.2.3.1.3

Multiply $- 8$ by $0$ .

$x = 0 + 0 - 22$

Step 1.2.3.2

Simplify by adding and subtracting.

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Step 1.2.3.2.1

Add $0$ and $0$ .

$x = 0 - 22$

Step 1.2.3.2.2

Subtract $22$ from $0$ .

$x = - 22$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- 22, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

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

$0 = - y^{2} - 8 y - 22$

Step 2.2

Solve the equation.

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Step 2.2.1

Rewrite the equation as $- y^{2} - 8 y - 22 = 0$ .

$- y^{2} - 8 y - 22 = 0$

Step 2.2.2

Use the quadratic formula to find the solutions.

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

Step 2.2.3

Substitute the values $a = - 1$ , $b = - 8$ , and $c = - 22$ into the quadratic formula and solve for $y$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (- 1 \cdot - 22)}}{2 \cdot - 1}$

Step 2.2.4

Simplify.

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Step 2.2.4.1

Simplify the numerator.

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Step 2.2.4.1.1

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

$y = \frac{8 \pm \sqrt{64 - 4 \cdot - 1 \cdot - 22}}{2 \cdot - 1}$

Step 2.2.4.1.2

Multiply $- 4 \cdot - 1 \cdot - 22$ .

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Step 2.2.4.1.2.1

Multiply $- 4$ by $- 1$ .

$y = \frac{8 \pm \sqrt{64 + 4 \cdot - 22}}{2 \cdot - 1}$

Step 2.2.4.1.2.2

Multiply $4$ by $- 22$ .

$y = \frac{8 \pm \sqrt{64 - 88}}{2 \cdot - 1}$

Step 2.2.4.1.3

Subtract $88$ from $64$ .

$y = \frac{8 \pm \sqrt{- 24}}{2 \cdot - 1}$

Step 2.2.4.1.4

Rewrite $- 24$ as $- 1 (24)$ .

$y = \frac{8 \pm \sqrt{- 1 \cdot 24}}{2 \cdot - 1}$

Step 2.2.4.1.5

Rewrite $\sqrt{- 1 (24)}$ as $\sqrt{- 1} \cdot \sqrt{24}$ .

$y = \frac{8 \pm \sqrt{- 1} \cdot \sqrt{24}}{2 \cdot - 1}$

Step 2.2.4.1.6

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

$y = \frac{8 \pm i \cdot \sqrt{24}}{2 \cdot - 1}$

Step 2.2.4.1.7

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

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Step 2.2.4.1.7.1

Factor $4$ out of $24$ .

$y = \frac{8 \pm i \cdot \sqrt{4 (6)}}{2 \cdot - 1}$

Step 2.2.4.1.7.2

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

$y = \frac{8 \pm i \cdot \sqrt{2^{2} \cdot 6}}{2 \cdot - 1}$

Step 2.2.4.1.8

Pull terms out from under the radical.

$y = \frac{8 \pm i \cdot (2 \sqrt{6})}{2 \cdot - 1}$

Step 2.2.4.1.9

Move $2$ to the left of $i$ .

$y = \frac{8 \pm 2 i \sqrt{6}}{2 \cdot - 1}$

Step 2.2.4.2

Multiply $2$ by $- 1$ .

$y = \frac{8 \pm 2 i \sqrt{6}}{- 2}$

Step 2.2.4.3

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

$y = \frac{4 \pm i \sqrt{6}}{- 1}$

Step 2.2.4.4

Move the negative one from the denominator of $\frac{4 \pm i \sqrt{6}}{- 1}$ .

$y = - 1 \cdot (4 \pm i \sqrt{6})$

Step 2.2.4.5

Rewrite $- 1 \cdot (4 \pm i \sqrt{6})$ as $- (4 \pm i \sqrt{6})$ .

$y = - (4 \pm i \sqrt{6})$

Step 2.2.5

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

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Step 2.2.5.1

Simplify the numerator.

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Step 2.2.5.1.1

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

$y = \frac{8 \pm \sqrt{64 - 4 \cdot - 1 \cdot - 22}}{2 \cdot - 1}$

Step 2.2.5.1.2

Multiply $- 4 \cdot - 1 \cdot - 22$ .

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Step 2.2.5.1.2.1

Multiply $- 4$ by $- 1$ .

$y = \frac{8 \pm \sqrt{64 + 4 \cdot - 22}}{2 \cdot - 1}$

Step 2.2.5.1.2.2

Multiply $4$ by $- 22$ .

$y = \frac{8 \pm \sqrt{64 - 88}}{2 \cdot - 1}$

Step 2.2.5.1.3

Subtract $88$ from $64$ .

$y = \frac{8 \pm \sqrt{- 24}}{2 \cdot - 1}$

Step 2.2.5.1.4

Rewrite $- 24$ as $- 1 (24)$ .

$y = \frac{8 \pm \sqrt{- 1 \cdot 24}}{2 \cdot - 1}$

Step 2.2.5.1.5

Rewrite $\sqrt{- 1 (24)}$ as $\sqrt{- 1} \cdot \sqrt{24}$ .

$y = \frac{8 \pm \sqrt{- 1} \cdot \sqrt{24}}{2 \cdot - 1}$

Step 2.2.5.1.6

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

$y = \frac{8 \pm i \cdot \sqrt{24}}{2 \cdot - 1}$

Step 2.2.5.1.7

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

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Step 2.2.5.1.7.1

Factor $4$ out of $24$ .

$y = \frac{8 \pm i \cdot \sqrt{4 (6)}}{2 \cdot - 1}$

Step 2.2.5.1.7.2

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

$y = \frac{8 \pm i \cdot \sqrt{2^{2} \cdot 6}}{2 \cdot - 1}$

Step 2.2.5.1.8

Pull terms out from under the radical.

$y = \frac{8 \pm i \cdot (2 \sqrt{6})}{2 \cdot - 1}$

Step 2.2.5.1.9

Move $2$ to the left of $i$ .

$y = \frac{8 \pm 2 i \sqrt{6}}{2 \cdot - 1}$

Step 2.2.5.2

Multiply $2$ by $- 1$ .

$y = \frac{8 \pm 2 i \sqrt{6}}{- 2}$

Step 2.2.5.3

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

$y = \frac{4 \pm i \sqrt{6}}{- 1}$

Step 2.2.5.4

Move the negative one from the denominator of $\frac{4 \pm i \sqrt{6}}{- 1}$ .

$y = - 1 \cdot (4 \pm i \sqrt{6})$

Step 2.2.5.5

Rewrite $- 1 \cdot (4 \pm i \sqrt{6})$ as $- (4 \pm i \sqrt{6})$ .

$y = - (4 \pm i \sqrt{6})$

Step 2.2.5.6

Change the $\pm$ to $+$ .

$y = - (4 + i \sqrt{6})$

Step 2.2.5.7

Apply the distributive property.

$y = - 1 \cdot 4 - (i \sqrt{6})$

Step 2.2.5.8

Multiply $- 1$ by $4$ .

$y = - 4 - i \sqrt{6}$

Step 2.2.6

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

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Step 2.2.6.1

Simplify the numerator.

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Step 2.2.6.1.1

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

$y = \frac{8 \pm \sqrt{64 - 4 \cdot - 1 \cdot - 22}}{2 \cdot - 1}$

Step 2.2.6.1.2

Multiply $- 4 \cdot - 1 \cdot - 22$ .

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Step 2.2.6.1.2.1

Multiply $- 4$ by $- 1$ .

$y = \frac{8 \pm \sqrt{64 + 4 \cdot - 22}}{2 \cdot - 1}$

Step 2.2.6.1.2.2

Multiply $4$ by $- 22$ .

$y = \frac{8 \pm \sqrt{64 - 88}}{2 \cdot - 1}$

Step 2.2.6.1.3

Subtract $88$ from $64$ .

$y = \frac{8 \pm \sqrt{- 24}}{2 \cdot - 1}$

Step 2.2.6.1.4

Rewrite $- 24$ as $- 1 (24)$ .

$y = \frac{8 \pm \sqrt{- 1 \cdot 24}}{2 \cdot - 1}$

Step 2.2.6.1.5

Rewrite $\sqrt{- 1 (24)}$ as $\sqrt{- 1} \cdot \sqrt{24}$ .

$y = \frac{8 \pm \sqrt{- 1} \cdot \sqrt{24}}{2 \cdot - 1}$

Step 2.2.6.1.6

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

$y = \frac{8 \pm i \cdot \sqrt{24}}{2 \cdot - 1}$

Step 2.2.6.1.7

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

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Step 2.2.6.1.7.1

Factor $4$ out of $24$ .

$y = \frac{8 \pm i \cdot \sqrt{4 (6)}}{2 \cdot - 1}$

Step 2.2.6.1.7.2

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

$y = \frac{8 \pm i \cdot \sqrt{2^{2} \cdot 6}}{2 \cdot - 1}$

Step 2.2.6.1.8

Pull terms out from under the radical.

$y = \frac{8 \pm i \cdot (2 \sqrt{6})}{2 \cdot - 1}$

Step 2.2.6.1.9

Move $2$ to the left of $i$ .

$y = \frac{8 \pm 2 i \sqrt{6}}{2 \cdot - 1}$

Step 2.2.6.2

Multiply $2$ by $- 1$ .

$y = \frac{8 \pm 2 i \sqrt{6}}{- 2}$

Step 2.2.6.3

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

$y = \frac{4 \pm i \sqrt{6}}{- 1}$

Step 2.2.6.4

Move the negative one from the denominator of $\frac{4 \pm i \sqrt{6}}{- 1}$ .

$y = - 1 \cdot (4 \pm i \sqrt{6})$

Step 2.2.6.5

Rewrite $- 1 \cdot (4 \pm i \sqrt{6})$ as $- (4 \pm i \sqrt{6})$ .

$y = - (4 \pm i \sqrt{6})$

Step 2.2.6.6

Change the $\pm$ to $-$ .

$y = - (4 - i \sqrt{6})$

Step 2.2.6.7

Apply the distributive property.

$y = - 1 \cdot 4 - (- i \sqrt{6})$

Step 2.2.6.8

Multiply $- 1$ by $4$ .

$y = - 4 - (- i \sqrt{6})$

Step 2.2.6.9

Multiply $- (- i \sqrt{6})$ .

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Step 2.2.6.9.1

Multiply $- 1$ by $- 1$ .

$y = - 4 + 1 (i \sqrt{6})$

Step 2.2.6.9.2

Multiply $\sqrt{6}$ by $1$ .

$y = - 4 + \sqrt{6} i$

Step 2.2.7

The final answer is the combination of both solutions.

$y = - 4 - i \sqrt{6}, - 4 + \sqrt{6} i$

Step 2.3

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

y-intercept(s): $None$

Step 3

List the intersections.

x-intercept(s): $(- 22, 0)$

y-intercept(s): $None$

Step 4