Precalculus Examples

${(x - 2)}^{2} + {(y + 3)}^{2} = 16$

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 $x$ .

${(x - 2)}^{2} + {((0) + 3)}^{2} = 16$

Step 1.2

Solve the equation.

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Step 1.2.1

Subtract ${((0) + 3)}^{2}$ from both sides of the equation.

${(x - 2)}^{2} = 16 - {((0) + 3)}^{2}$

Step 1.2.2

Add $0$ and $3$ .

${(x - 2)}^{2} = 16 - 3^{2}$

Step 1.2.3

Raise $3$ to the power of $2$ .

${(x - 2)}^{2} = 16 - 1 \cdot 9$

Step 1.2.4

Multiply $- 1$ by $9$ .

${(x - 2)}^{2} = 16 - 9$

Step 1.2.5

Subtract $9$ from $16$ .

${(x - 2)}^{2} = 7$

Step 1.2.6

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

$x - 2 = \pm \sqrt{7}$

Step 1.2.7

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

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Step 1.2.7.1

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

$x - 2 = \sqrt{7}$

Step 1.2.7.2

Add $2$ to both sides of the equation.

$x = \sqrt{7} + 2$

Step 1.2.7.3

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

$x - 2 = - \sqrt{7}$

Step 1.2.7.4

Add $2$ to both sides of the equation.

$x = - \sqrt{7} + 2$

Step 1.2.7.5

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

$x = \sqrt{7} + 2, - \sqrt{7} + 2$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\sqrt{7} + 2, 0), (- \sqrt{7} + 2, 0)$

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 $y$ .

${((0) - 2)}^{2} + {(y + 3)}^{2} = 16$

Step 2.2

Solve the equation.

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Step 2.2.1

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

${(y + 3)}^{2} = 16 - {((0) - 2)}^{2}$

Step 2.2.2

Subtract $2$ from $0$ .

${(y + 3)}^{2} = 16 - {(- 2)}^{2}$

Step 2.2.3

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

${(y + 3)}^{2} = 16 - 1 \cdot 4$

Step 2.2.4

Multiply $- 1$ by $4$ .

${(y + 3)}^{2} = 16 - 4$

Step 2.2.5

Subtract $4$ from $16$ .

${(y + 3)}^{2} = 12$

Step 2.2.6

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

$y + 3 = \pm \sqrt{12}$

Step 2.2.7

Simplify $\pm \sqrt{12}$ .

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Step 2.2.7.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.2.7.1.1

Factor $4$ out of $12$ .

$y + 3 = \pm \sqrt{4 (3)}$

Step 2.2.7.1.2

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

$y + 3 = \pm \sqrt{2^{2} \cdot 3}$

Step 2.2.7.2

Pull terms out from under the radical.

$y + 3 = \pm 2 \sqrt{3}$

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.

$y + 3 = 2 \sqrt{3}$

Step 2.2.8.2

Subtract $3$ from both sides of the equation.

$y = 2 \sqrt{3} - 3$

Step 2.2.8.3

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

$y + 3 = - 2 \sqrt{3}$

Step 2.2.8.4

Subtract $3$ from both sides of the equation.

$y = - 2 \sqrt{3} - 3$

Step 2.2.8.5

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

$y = 2 \sqrt{3} - 3, - 2 \sqrt{3} - 3$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 2 \sqrt{3} - 3), (0, - 2 \sqrt{3} - 3)$

Step 3

List the intersections.

x-intercept(s): $(\sqrt{7} + 2, 0), (- \sqrt{7} + 2, 0)$

y-intercept(s): $(0, 2 \sqrt{3} - 3), (0, - 2 \sqrt{3} - 3)$

Step 4