Precalculus Examples

$x^{2} + {(y - k)}^{2} = 21$

Step 1

Subtract $x^{2}$ from both sides of the equation.

${(y - k)}^{2} = 21 - x^{2}$

Step 2

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

$y - k = \pm \sqrt{21 - x^{2}}$

Step 3

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

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Step 3.1

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

$y - k = \sqrt{21 - x^{2}}$

Step 3.2

Subtract $y$ from both sides of the equation.

$- k = \sqrt{21 - x^{2}} - y$

Step 3.3

Divide each term in $- k = \sqrt{21 - x^{2}} - y$ by $- 1$ and simplify.

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Step 3.3.1

Divide each term in $- k = \sqrt{21 - x^{2}} - y$ by $- 1$ .

$\frac{- k}{- 1} = \frac{\sqrt{21 - x^{2}}}{- 1} + \frac{- y}{- 1}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Dividing two negative values results in a positive value.

$\frac{k}{1} = \frac{\sqrt{21 - x^{2}}}{- 1} + \frac{- y}{- 1}$

Step 3.3.2.2

Divide $k$ by $1$ .

$k = \frac{\sqrt{21 - x^{2}}}{- 1} + \frac{- y}{- 1}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

Move the negative one from the denominator of $\frac{\sqrt{21 - x^{2}}}{- 1}$ .

$k = - 1 \cdot \sqrt{21 - x^{2}} + \frac{- y}{- 1}$

Step 3.3.3.1.2

Rewrite $- 1 \cdot \sqrt{21 - x^{2}}$ as $- \sqrt{21 - x^{2}}$ .

$k = - \sqrt{21 - x^{2}} + \frac{- y}{- 1}$

Step 3.3.3.1.3

Dividing two negative values results in a positive value.

$k = - \sqrt{21 - x^{2}} + \frac{y}{1}$

Step 3.3.3.1.4

Divide $y$ by $1$ .

$k = - \sqrt{21 - x^{2}} + y$

Step 3.4

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

$y - k = - \sqrt{21 - x^{2}}$

Step 3.5

Subtract $y$ from both sides of the equation.

$- k = - \sqrt{21 - x^{2}} - y$

Step 3.6

Divide each term in $- k = - \sqrt{21 - x^{2}} - y$ by $- 1$ and simplify.

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Step 3.6.1

Divide each term in $- k = - \sqrt{21 - x^{2}} - y$ by $- 1$ .

$\frac{- k}{- 1} = \frac{- \sqrt{21 - x^{2}}}{- 1} + \frac{- y}{- 1}$

Step 3.6.2

Simplify the left side.

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Step 3.6.2.1

Dividing two negative values results in a positive value.

$\frac{k}{1} = \frac{- \sqrt{21 - x^{2}}}{- 1} + \frac{- y}{- 1}$

Step 3.6.2.2

Divide $k$ by $1$ .

$k = \frac{- \sqrt{21 - x^{2}}}{- 1} + \frac{- y}{- 1}$

Step 3.6.3

Simplify the right side.

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Step 3.6.3.1

Simplify each term.

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Step 3.6.3.1.1

Dividing two negative values results in a positive value.

$k = \frac{\sqrt{21 - x^{2}}}{1} + \frac{- y}{- 1}$

Step 3.6.3.1.2

Divide $\sqrt{21 - x^{2}}$ by $1$ .

$k = \sqrt{21 - x^{2}} + \frac{- y}{- 1}$

Step 3.6.3.1.3

Dividing two negative values results in a positive value.

$k = \sqrt{21 - x^{2}} + \frac{y}{1}$

Step 3.6.3.1.4

Divide $y$ by $1$ .

$k = \sqrt{21 - x^{2}} + y$

Step 3.7

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

$k = - \sqrt{21 - x^{2}} + y$

$k = \sqrt{21 - x^{2}} + y$

$k = - \sqrt{21 - x^{2}} + y$

$k = \sqrt{21 - x^{2}} + y$