Precalculus Examples

${(x - 1)}^{2} + {(y - 2)}^{2} = 25$

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 $x$ .

${(x - 1)}^{2} + {((0) - 2)}^{2} = 25$

Step 1.2

Solve the equation.

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Step 1.2.1

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

${(x - 1)}^{2} = 25 - {((0) - 2)}^{2}$

Step 1.2.2

Subtract $2$ from $0$ .

${(x - 1)}^{2} = 25 - {(- 2)}^{2}$

Step 1.2.3

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

${(x - 1)}^{2} = 25 - 1 \cdot 4$

Step 1.2.4

Multiply $- 1$ by $4$ .

${(x - 1)}^{2} = 25 - 4$

Step 1.2.5

Subtract $4$ from $25$ .

${(x - 1)}^{2} = 21$

Step 1.2.6

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

$x - 1 = \pm \sqrt{21}$

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.

$x - 1 = \sqrt{21}$

Step 1.2.7.2

Add $1$ to both sides of the equation.

$x = \sqrt{21} + 1$

Step 1.2.7.3

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

$x - 1 = - \sqrt{21}$

Step 1.2.7.4

Add $1$ to both sides of the equation.

$x = - \sqrt{21} + 1$

Step 1.2.7.5

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

$x = \sqrt{21} + 1, - \sqrt{21} + 1$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\sqrt{21} + 1, 0), (- \sqrt{21} + 1, 0)$

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 $y$ .

${((0) - 1)}^{2} + {(y - 2)}^{2} = 25$

Step 2.2

Solve the equation.

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Step 2.2.1

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

${(y - 2)}^{2} = 25 - {((0) - 1)}^{2}$

Step 2.2.2

Subtract $1$ from $0$ .

${(y - 2)}^{2} = 25 - {(- 1)}^{2}$

Step 2.2.3

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

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Step 2.2.3.1

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

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Step 2.2.3.1.1

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

${(y - 2)}^{2} = 25 + {(- 1)}^{1} {(- 1)}^{2}$

Step 2.2.3.1.2

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

${(y - 2)}^{2} = 25 + {(- 1)}^{1 + 2}$

Step 2.2.3.2

Add $1$ and $2$ .

${(y - 2)}^{2} = 25 + {(- 1)}^{3}$

Step 2.2.4

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

$y - 2 = \pm \sqrt{25 + {(- 1)}^{3}}$

Step 2.2.5

Simplify $\pm \sqrt{25 + {(- 1)}^{3}}$ .

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Step 2.2.5.1

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

$y - 2 = \pm \sqrt{25 - 1}$

Step 2.2.5.2

Subtract $1$ from $25$ .

$y - 2 = \pm \sqrt{24}$

Step 2.2.5.3

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

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Step 2.2.5.3.1

Factor $4$ out of $24$ .

$y - 2 = \pm \sqrt{4 (6)}$

Step 2.2.5.3.2

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

$y - 2 = \pm \sqrt{2^{2} \cdot 6}$

Step 2.2.5.4

Pull terms out from under the radical.

$y - 2 = \pm 2 \sqrt{6}$

Step 2.2.6

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

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Step 2.2.6.1

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

$y - 2 = 2 \sqrt{6}$

Step 2.2.6.2

Add $2$ to both sides of the equation.

$y = 2 \sqrt{6} + 2$

Step 2.2.6.3

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

$y - 2 = - 2 \sqrt{6}$

Step 2.2.6.4

Add $2$ to both sides of the equation.

$y = - 2 \sqrt{6} + 2$

Step 2.2.6.5

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

$y = 2 \sqrt{6} + 2, - 2 \sqrt{6} + 2$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 2 \sqrt{6} + 2), (0, - 2 \sqrt{6} + 2)$

Step 3

List the intersections.

x-intercept(s): $(\sqrt{21} + 1, 0), (- \sqrt{21} + 1, 0)$

y-intercept(s): $(0, 2 \sqrt{6} + 2), (0, - 2 \sqrt{6} + 2)$

Step 4