Precalculus Examples

${(x + 5)}^{2} + {(y + 4)}^{2} = 64$

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 $x$ .

${(x + 5)}^{2} + {((0) + 4)}^{2} = 64$

Step 1.2

Solve the equation.

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Step 1.2.1

Subtract ${((0) + 4)}^{2}$ from both sides of the equation.

${(x + 5)}^{2} = 64 - {((0) + 4)}^{2}$

Step 1.2.2

Add $0$ and $4$ .

${(x + 5)}^{2} = 64 - 4^{2}$

Step 1.2.3

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

${(x + 5)}^{2} = 64 - 1 \cdot 16$

Step 1.2.4

Multiply $- 1$ by $16$ .

${(x + 5)}^{2} = 64 - 16$

Step 1.2.5

Subtract $16$ from $64$ .

${(x + 5)}^{2} = 48$

Step 1.2.6

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

$x + 5 = \pm \sqrt{48}$

Step 1.2.7

Simplify $\pm \sqrt{48}$ .

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Step 1.2.7.1

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

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Step 1.2.7.1.1

Factor $16$ out of $48$ .

$x + 5 = \pm \sqrt{16 (3)}$

Step 1.2.7.1.2

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

$x + 5 = \pm \sqrt{4^{2} \cdot 3}$

Step 1.2.7.2

Pull terms out from under the radical.

$x + 5 = \pm 4 \sqrt{3}$

Step 1.2.8

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

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Step 1.2.8.1

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

$x + 5 = 4 \sqrt{3}$

Step 1.2.8.2

Subtract $5$ from both sides of the equation.

$x = 4 \sqrt{3} - 5$

Step 1.2.8.3

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

$x + 5 = - 4 \sqrt{3}$

Step 1.2.8.4

Subtract $5$ from both sides of the equation.

$x = - 4 \sqrt{3} - 5$

Step 1.2.8.5

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

$x = 4 \sqrt{3} - 5, - 4 \sqrt{3} - 5$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(4 \sqrt{3} - 5, 0), (- 4 \sqrt{3} - 5, 0)$

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 $y$ .

${((0) + 5)}^{2} + {(y + 4)}^{2} = 64$

Step 2.2

Solve the equation.

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Step 2.2.1

Subtract ${((0) + 5)}^{2}$ from both sides of the equation.

${(y + 4)}^{2} = 64 - {((0) + 5)}^{2}$

Step 2.2.2

Add $0$ and $5$ .

${(y + 4)}^{2} = 64 - 5^{2}$

Step 2.2.3

Raise $5$ to the power of $2$ .

${(y + 4)}^{2} = 64 - 1 \cdot 25$

Step 2.2.4

Multiply $- 1$ by $25$ .

${(y + 4)}^{2} = 64 - 25$

Step 2.2.5

Subtract $25$ from $64$ .

${(y + 4)}^{2} = 39$

Step 2.2.6

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

$y + 4 = \pm \sqrt{39}$

Step 2.2.7

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

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Step 2.2.7.1

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

$y + 4 = \sqrt{39}$

Step 2.2.7.2

Subtract $4$ from both sides of the equation.

$y = \sqrt{39} - 4$

Step 2.2.7.3

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

$y + 4 = - \sqrt{39}$

Step 2.2.7.4

Subtract $4$ from both sides of the equation.

$y = - \sqrt{39} - 4$

Step 2.2.7.5

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

$y = \sqrt{39} - 4, - \sqrt{39} - 4$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, \sqrt{39} - 4), (0, - \sqrt{39} - 4)$

Step 3

List the intersections.

x-intercept(s): $(4 \sqrt{3} - 5, 0), (- 4 \sqrt{3} - 5, 0)$

y-intercept(s): $(0, \sqrt{39} - 4), (0, - \sqrt{39} - 4)$

Step 4