Precalculus Examples

${(x + 8)}^{2} + {(y - 1)}^{2} = 7$

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 $x$ .

${(x + 8)}^{2} + {((0) - 1)}^{2} = 7$

Step 1.2

Solve the equation.

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Step 1.2.1

Subtract ${((0) - 1)}^{2}$ from both sides of the equation.

${(x + 8)}^{2} = 7 - {((0) - 1)}^{2}$

Step 1.2.2

Subtract $1$ from $0$ .

${(x + 8)}^{2} = 7 - {(- 1)}^{2}$

Step 1.2.3

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 1.2.3.1

Multiply $- 1$ by ${(- 1)}^{2}$ .

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Step 1.2.3.1.1

Raise $- 1$ to the power of $1$ .

${(x + 8)}^{2} = 7 + {(- 1)}^{1} {(- 1)}^{2}$

Step 1.2.3.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(x + 8)}^{2} = 7 + {(- 1)}^{1 + 2}$

Step 1.2.3.2

Add $1$ and $2$ .

${(x + 8)}^{2} = 7 + {(- 1)}^{3}$

Step 1.2.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x + 8 = \pm \sqrt{7 + {(- 1)}^{3}}$

Step 1.2.5

Simplify $\pm \sqrt{7 + {(- 1)}^{3}}$ .

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Step 1.2.5.1

Raise $- 1$ to the power of $3$ .

$x + 8 = \pm \sqrt{7 - 1}$

Step 1.2.5.2

Subtract $1$ from $7$ .

$x + 8 = \pm \sqrt{6}$

Step 1.2.6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.6.1

First, use the positive value of the $\pm$ to find the first solution.

$x + 8 = \sqrt{6}$

Step 1.2.6.2

Subtract $8$ from both sides of the equation.

$x = \sqrt{6} - 8$

Step 1.2.6.3

Next, use the negative value of the $\pm$ to find the second solution.

$x + 8 = - \sqrt{6}$

Step 1.2.6.4

Subtract $8$ from both sides of the equation.

$x = - \sqrt{6} - 8$

Step 1.2.6.5

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{6} - 8, - \sqrt{6} - 8$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\sqrt{6} - 8, 0), (- \sqrt{6} - 8, 0)$

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 $y$ .

${((0) + 8)}^{2} + {(y - 1)}^{2} = 7$

Step 2.2

Solve the equation.

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Step 2.2.1

Subtract ${((0) + 8)}^{2}$ from both sides of the equation.

${(y - 1)}^{2} = 7 - {((0) + 8)}^{2}$

Step 2.2.2

Add $0$ and $8$ .

${(y - 1)}^{2} = 7 - 8^{2}$

Step 2.2.3

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

${(y - 1)}^{2} = 7 - 1 \cdot 64$

Step 2.2.4

Multiply $- 1$ by $64$ .

${(y - 1)}^{2} = 7 - 64$

Step 2.2.5

Subtract $64$ from $7$ .

${(y - 1)}^{2} = - 57$

Step 2.2.6

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y - 1 = \pm \sqrt{- 57}$

Step 2.2.7

Simplify $\pm \sqrt{- 57}$ .

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Step 2.2.7.1

Rewrite $- 57$ as $- 1 (57)$ .

$y - 1 = \pm \sqrt{- 1 (57)}$

Step 2.2.7.2

Rewrite $\sqrt{- 1 (57)}$ as $\sqrt{- 1} \cdot \sqrt{57}$ .

$y - 1 = \pm \sqrt{- 1} \cdot \sqrt{57}$

Step 2.2.7.3

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

$y - 1 = \pm i \sqrt{57}$

Step 2.2.8

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.2.8.1

First, use the positive value of the $\pm$ to find the first solution.

$y - 1 = i \sqrt{57}$

Step 2.2.8.2

Add $1$ to both sides of the equation.

$y = i \sqrt{57} + 1$

Step 2.2.8.3

Next, use the negative value of the $\pm$ to find the second solution.

$y - 1 = - i \sqrt{57}$

Step 2.2.8.4

Add $1$ to both sides of the equation.

$y = - i \sqrt{57} + 1$

Step 2.2.8.5

The complete solution is the result of both the positive and negative portions of the solution.

$y = i \sqrt{57} + 1, - i \sqrt{57} + 1$

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): $(\sqrt{6} - 8, 0), (- \sqrt{6} - 8, 0)$

y-intercept(s): $None$

Step 4