Algebra Examples

${(x - 2)}^{2} + y^{2} = 13$

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 $x$ .

${(x - 2)}^{2} + {(0)}^{2} = 13$

Step 1.2

Solve the equation.

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Step 1.2.1

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

${(x - 2)}^{2} = 13 - {(0)}^{2}$

Step 1.2.2

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

${(x - 2)}^{2} = 13 - 0$

Step 1.2.3

Multiply $- 1$ by $0$ .

${(x - 2)}^{2} = 13 + 0$

Step 1.2.4

Add $13$ and $0$ .

${(x - 2)}^{2} = 13$

Step 1.2.5

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

$x - 2 = \pm \sqrt{13}$

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.

$x - 2 = \sqrt{13}$

Step 1.2.6.2

Add $2$ to both sides of the equation.

$x = \sqrt{13} + 2$

Step 1.2.6.3

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

$x - 2 = - \sqrt{13}$

Step 1.2.6.4

Add $2$ to both sides of the equation.

$x = - \sqrt{13} + 2$

Step 1.2.6.5

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

$x = \sqrt{13} + 2, - \sqrt{13} + 2$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\sqrt{13} + 2, 0), (- \sqrt{13} + 2, 0)$

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 $y$ .

${((0) - 2)}^{2} + y^{2} = 13$

Step 2.2

Solve the equation.

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Step 2.2.1

Simplify ${((0) - 2)}^{2} + y^{2}$ .

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Step 2.2.1.1

Subtract $2$ from $0$ .

${(- 2)}^{2} + y^{2} = 13$

Step 2.2.1.2

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

$4 + y^{2} = 13$

Step 2.2.2

Move all terms not containing $y$ to the right side of the equation.

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Step 2.2.2.1

Subtract $4$ from both sides of the equation.

$y^{2} = 13 - 4$

Step 2.2.2.2

Subtract $4$ from $13$ .

$y^{2} = 9$

Step 2.2.3

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

$y = \pm \sqrt{9}$

Step 2.2.4

Simplify $\pm \sqrt{9}$ .

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Step 2.2.4.1

Rewrite $9$ as $3^{2}$ .

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

Step 2.2.4.2

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

$y = \pm 3$

Step 2.2.5

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

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Step 2.2.5.1

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

$y = 3$

Step 2.2.5.2

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

$y = - 3$

Step 2.2.5.3

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

$y = 3, - 3$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 3), (0, - 3)$

Step 3

List the intersections.

x-intercept(s): $(\sqrt{13} + 2, 0), (- \sqrt{13} + 2, 0)$

y-intercept(s): $(0, 3), (0, - 3)$

Step 4