Finite Math Examples

$r = {(x - (- 3))}^{2} + {(y - 6)}^{2}$

Step 1

Rewrite the equation as ${(x - (- 3))}^{2} + {(y - 6)}^{2} = r$ .

${(x - (- 3))}^{2} + {(y - 6)}^{2} = r$

Step 2

Subtract ${(y - 6)}^{2}$ from both sides of the equation.

${(x - (- 3))}^{2} = r - {(y - 6)}^{2}$

Step 3

Multiply $- 1$ by $- 3$ .

${(x + 3)}^{2} = r - {(y - 6)}^{2}$

Step 4

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

$x + 3 = \pm \sqrt{r - {(y - 6)}^{2}}$

Step 5

Simplify $\pm \sqrt{r - {(y - 6)}^{2}}$ .

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Step 5.1

Rewrite ${(y - 6)}^{2}$ as $(y - 6) (y - 6)$ .

$x + 3 = \pm \sqrt{r - ((y - 6) (y - 6))}$

Step 5.2

Expand $(y - 6) (y - 6)$ using the FOIL Method.

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Step 5.2.1

Apply the distributive property.

$x + 3 = \pm \sqrt{r - (y (y - 6) - 6 (y - 6))}$

Step 5.2.2

Apply the distributive property.

$x + 3 = \pm \sqrt{r - (y \cdot y + y \cdot - 6 - 6 (y - 6))}$

Step 5.2.3

Apply the distributive property.

$x + 3 = \pm \sqrt{r - (y \cdot y + y \cdot - 6 - 6 y - 6 \cdot - 6)}$

Step 5.3

Simplify and combine like terms.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Multiply $y$ by $y$ .

$x + 3 = \pm \sqrt{r - (y^{2} + y \cdot - 6 - 6 y - 6 \cdot - 6)}$

Step 5.3.1.2

Move $- 6$ to the left of $y$ .

$x + 3 = \pm \sqrt{r - (y^{2} - 6 \cdot y - 6 y - 6 \cdot - 6)}$

Step 5.3.1.3

Multiply $- 6$ by $- 6$ .

$x + 3 = \pm \sqrt{r - (y^{2} - 6 y - 6 y + 36)}$

Step 5.3.2

Subtract $6 y$ from $- 6 y$ .

$x + 3 = \pm \sqrt{r - (y^{2} - 12 y + 36)}$

Step 5.4

Apply the distributive property.

$x + 3 = \pm \sqrt{r - y^{2} - (- 12 y) - 1 \cdot 36}$

Step 5.5

Simplify.

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Step 5.5.1

Multiply $- 12$ by $- 1$ .

$x + 3 = \pm \sqrt{r - y^{2} + 12 y - 1 \cdot 36}$

Step 5.5.2

Multiply $- 1$ by $36$ .

$x + 3 = \pm \sqrt{r - y^{2} + 12 y - 36}$

Step 5.6

Rewrite $r - y^{2} + 12 y - 36$ in a factored form.

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Step 5.6.1

Regroup terms.

$x + 3 = \pm \sqrt{- 36 + r - y^{2} + 12 y}$

Step 5.6.2

Factor $- 1$ out of $r - y^{2} + 12 y$ .

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Step 5.6.2.1

Reorder $r$ and $- y^{2}$ .

$x + 3 = \pm \sqrt{- 36 - y^{2} + r + 12 y}$

Step 5.6.2.2

Factor $- 1$ out of $- y^{2}$ .

$x + 3 = \pm \sqrt{- 36 - (y^{2}) + r + 12 y}$

Step 5.6.2.3

Factor $- 1$ out of $r$ .

$x + 3 = \pm \sqrt{- 36 - (y^{2}) - 1 (- r) + 12 y}$

Step 5.6.2.4

Factor $- 1$ out of $12 y$ .

$x + 3 = \pm \sqrt{- 36 - (y^{2}) - 1 (- r) - (- 12 y)}$

Step 5.6.2.5

Factor $- 1$ out of $- (y^{2}) - 1 (- r)$ .

$x + 3 = \pm \sqrt{- 36 - (y^{2} - r) - (- 12 y)}$

Step 5.6.2.6

Factor $- 1$ out of $- (y^{2} - r) - (- 12 y)$ .

$x + 3 = \pm \sqrt{- 36 - (y^{2} - r - 12 y)}$

Step 5.6.3

Factor $- 1$ out of $- 36 - (y^{2} - r - 12 y)$ .

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Step 5.6.3.1

Reorder $- 36$ and $- (y^{2} - r - 12 y)$ .

$x + 3 = \pm \sqrt{- (y^{2} - r - 12 y) - 36}$

Step 5.6.3.2

Rewrite $- 36$ as $- 1 (36)$ .

$x + 3 = \pm \sqrt{- (y^{2} - r - 12 y) - 1 (36)}$

Step 5.6.3.3

Factor $- 1$ out of $- (y^{2} - r - 12 y) - 1 (36)$ .

$x + 3 = \pm \sqrt{- (y^{2} - r - 12 y + 36)}$

Step 6

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

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Step 6.1

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

$x + 3 = \sqrt{- (y^{2} - r - 12 y + 36)}$

Step 6.2

Subtract $3$ from both sides of the equation.

$x = \sqrt{- (y^{2} - r - 12 y + 36)} - 3$

Step 6.3

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

$x + 3 = - \sqrt{- (y^{2} - r - 12 y + 36)}$

Step 6.4

Subtract $3$ from both sides of the equation.

$x = - \sqrt{- (y^{2} - r - 12 y + 36)} - 3$

Step 6.5

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

$x = \sqrt{- (y^{2} - r - 12 y + 36)} - 3$

$x = - \sqrt{- (y^{2} - r - 12 y + 36)} - 3$

$x = \sqrt{- (y^{2} - r - 12 y + 36)} - 3$

$x = - \sqrt{- (y^{2} - r - 12 y + 36)} - 3$