Examples

{(x - 1)}^{2} + \frac{{(y + 4)}^{2}}{6} = 5

${(x - 1)}^{2} + \frac{{(y + 4)}^{2}}{6} = 5$

Step 1

Simplify the left side

{(x - 1)}^{2} + \frac{{(y + 4)}^{2}}{6}

${(x - 1)}^{2} + \frac{{(y + 4)}^{2}}{6}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite

{(x - 1)}^{2}

${(x - 1)}^{2}$ as

(x - 1) (x - 1)

$(x - 1) (x - 1)$ .

(x - 1) (x - 1) + \frac{{(y + 4)}^{2}}{6} = 5

$(x - 1) (x - 1) + \frac{{(y + 4)}^{2}}{6} = 5$

Step 1.1.2

Expand

(x - 1) (x - 1)

$(x - 1) (x - 1)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

x (x - 1) - 1 (x - 1) + \frac{{(y + 4)}^{2}}{6} = 5

$x (x - 1) - 1 (x - 1) + \frac{{(y + 4)}^{2}}{6} = 5$

Step 1.1.2.2

Apply the distributive property.

x \cdot x + x \cdot - 1 - 1 (x - 1) + \frac{{(y + 4)}^{2}}{6} = 5

$x \cdot x + x \cdot - 1 - 1 (x - 1) + \frac{{(y + 4)}^{2}}{6} = 5$

Step 1.1.2.3

Apply the distributive property.

x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1 + \frac{{(y + 4)}^{2}}{6} = 5

$x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1 + \frac{{(y + 4)}^{2}}{6} = 5$

x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1 + \frac{{(y + 4)}^{2}}{6} = 5

$x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1 + \frac{{(y + 4)}^{2}}{6} = 5$

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Multiply

x

$x$ by

x

$x$ .

x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1 + \frac{{(y + 4)}^{2}}{6} = 5

$x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1 + \frac{{(y + 4)}^{2}}{6} = 5$

Step 1.1.3.1.2

Move

- 1

$- 1$ to the left of

x

$x$ .

x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1 + \frac{{(y + 4)}^{2}}{6} = 5

$x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1 + \frac{{(y + 4)}^{2}}{6} = 5$

Step 1.1.3.1.3

Rewrite

- 1 x

$- 1 x$ as

- x

$- x$ .

x^{2} - x - 1 x - 1 \cdot - 1 + \frac{{(y + 4)}^{2}}{6} = 5

$x^{2} - x - 1 x - 1 \cdot - 1 + \frac{{(y + 4)}^{2}}{6} = 5$

Step 1.1.3.1.4

Rewrite

- 1 x

$- 1 x$ as

- x

$- x$ .

x^{2} - x - x - 1 \cdot - 1 + \frac{{(y + 4)}^{2}}{6} = 5

$x^{2} - x - x - 1 \cdot - 1 + \frac{{(y + 4)}^{2}}{6} = 5$

Step 1.1.3.1.5

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

x^{2} - x - x + 1 + \frac{{(y + 4)}^{2}}{6} = 5

$x^{2} - x - x + 1 + \frac{{(y + 4)}^{2}}{6} = 5$

x^{2} - x - x + 1 + \frac{{(y + 4)}^{2}}{6} = 5

$x^{2} - x - x + 1 + \frac{{(y + 4)}^{2}}{6} = 5$

Step 1.1.3.2

Subtract

x

$x$ from

- x

$- x$ .

x^{2} - 2 x + 1 + \frac{{(y + 4)}^{2}}{6} = 5

$x^{2} - 2 x + 1 + \frac{{(y + 4)}^{2}}{6} = 5$

x^{2} - 2 x + 1 + \frac{{(y + 4)}^{2}}{6} = 5

$x^{2} - 2 x + 1 + \frac{{(y + 4)}^{2}}{6} = 5$

x^{2} - 2 x + 1 + \frac{{(y + 4)}^{2}}{6} = 5

$x^{2} - 2 x + 1 + \frac{{(y + 4)}^{2}}{6} = 5$

Step 1.2

To write

x^{2}

$x^{2}$ as a fraction with a common denominator, multiply by

\frac{6}{6}

$\frac{6}{6}$ .

- 2 x + 1 + x^{2} \cdot \frac{6}{6} + \frac{{(y + 4)}^{2}}{6} = 5

$- 2 x + 1 + x^{2} \cdot \frac{6}{6} + \frac{{(y + 4)}^{2}}{6} = 5$

Step 1.3

Simplify terms.

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Step 1.3.1

Combine

x^{2}

$x^{2}$ and

\frac{6}{6}

$\frac{6}{6}$ .

- 2 x + 1 + \frac{x^{2} \cdot 6}{6} + \frac{{(y + 4)}^{2}}{6} = 5

$- 2 x + 1 + \frac{x^{2} \cdot 6}{6} + \frac{{(y + 4)}^{2}}{6} = 5$

Step 1.3.2

Combine the numerators over the common denominator.

- 2 x + 1 + \frac{x^{2} \cdot 6 + {(y + 4)}^{2}}{6} = 5

$- 2 x + 1 + \frac{x^{2} \cdot 6 + {(y + 4)}^{2}}{6} = 5$

- 2 x + 1 + \frac{x^{2} \cdot 6 + {(y + 4)}^{2}}{6} = 5

$- 2 x + 1 + \frac{x^{2} \cdot 6 + {(y + 4)}^{2}}{6} = 5$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Move

6

$6$ to the left of

x^{2}

$x^{2}$ .

- 2 x + 1 + \frac{6 \cdot x^{2} + {(y + 4)}^{2}}{6} = 5

$- 2 x + 1 + \frac{6 \cdot x^{2} + {(y + 4)}^{2}}{6} = 5$

Step 1.4.2

Rewrite

{(y + 4)}^{2}

${(y + 4)}^{2}$ as

(y + 4) (y + 4)

$(y + 4) (y + 4)$ .

- 2 x + 1 + \frac{6 x^{2} + (y + 4) (y + 4)}{6} = 5

$- 2 x + 1 + \frac{6 x^{2} + (y + 4) (y + 4)}{6} = 5$

Step 1.4.3

Expand

(y + 4) (y + 4)

$(y + 4) (y + 4)$ using the FOIL Method.

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Step 1.4.3.1

Apply the distributive property.

- 2 x + 1 + \frac{6 x^{2} + y (y + 4) + 4 (y + 4)}{6} = 5

$- 2 x + 1 + \frac{6 x^{2} + y (y + 4) + 4 (y + 4)}{6} = 5$

Step 1.4.3.2

Apply the distributive property.

- 2 x + 1 + \frac{6 x^{2} + y \cdot y + y \cdot 4 + 4 (y + 4)}{6} = 5

$- 2 x + 1 + \frac{6 x^{2} + y \cdot y + y \cdot 4 + 4 (y + 4)}{6} = 5$

Step 1.4.3.3

Apply the distributive property.

- 2 x + 1 + \frac{6 x^{2} + y \cdot y + y \cdot 4 + 4 y + 4 \cdot 4}{6} = 5

$- 2 x + 1 + \frac{6 x^{2} + y \cdot y + y \cdot 4 + 4 y + 4 \cdot 4}{6} = 5$

- 2 x + 1 + \frac{6 x^{2} + y \cdot y + y \cdot 4 + 4 y + 4 \cdot 4}{6} = 5

$- 2 x + 1 + \frac{6 x^{2} + y \cdot y + y \cdot 4 + 4 y + 4 \cdot 4}{6} = 5$

Step 1.4.4

Simplify and combine like terms.

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Step 1.4.4.1

Simplify each term.

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Step 1.4.4.1.1

Multiply

y

$y$ by

y

$y$ .

- 2 x + 1 + \frac{6 x^{2} + y^{2} + y \cdot 4 + 4 y + 4 \cdot 4}{6} = 5

$- 2 x + 1 + \frac{6 x^{2} + y^{2} + y \cdot 4 + 4 y + 4 \cdot 4}{6} = 5$

Step 1.4.4.1.2

Move

4

$4$ to the left of

y

$y$ .

- 2 x + 1 + \frac{6 x^{2} + y^{2} + 4 \cdot y + 4 y + 4 \cdot 4}{6} = 5

$- 2 x + 1 + \frac{6 x^{2} + y^{2} + 4 \cdot y + 4 y + 4 \cdot 4}{6} = 5$

Step 1.4.4.1.3

Multiply

4

$4$ by

4

$4$ .

- 2 x + 1 + \frac{6 x^{2} + y^{2} + 4 y + 4 y + 16}{6} = 5

$- 2 x + 1 + \frac{6 x^{2} + y^{2} + 4 y + 4 y + 16}{6} = 5$

- 2 x + 1 + \frac{6 x^{2} + y^{2} + 4 y + 4 y + 16}{6} = 5

$- 2 x + 1 + \frac{6 x^{2} + y^{2} + 4 y + 4 y + 16}{6} = 5$

Step 1.4.4.2

Add

4 y

$4 y$ and

4 y

$4 y$ .

- 2 x + 1 + \frac{6 x^{2} + y^{2} + 8 y + 16}{6} = 5

$- 2 x + 1 + \frac{6 x^{2} + y^{2} + 8 y + 16}{6} = 5$

- 2 x + 1 + \frac{6 x^{2} + y^{2} + 8 y + 16}{6} = 5

$- 2 x + 1 + \frac{6 x^{2} + y^{2} + 8 y + 16}{6} = 5$

- 2 x + 1 + \frac{6 x^{2} + y^{2} + 8 y + 16}{6} = 5

$- 2 x + 1 + \frac{6 x^{2} + y^{2} + 8 y + 16}{6} = 5$

Step 1.5

To write

- 2 x

$- 2 x$ as a fraction with a common denominator, multiply by

\frac{6}{6}

$\frac{6}{6}$ .

- 2 x \cdot \frac{6}{6} + \frac{6 x^{2} + y^{2} + 8 y + 16}{6} + 1 = 5

$- 2 x \cdot \frac{6}{6} + \frac{6 x^{2} + y^{2} + 8 y + 16}{6} + 1 = 5$

Step 1.6

Simplify terms.

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Step 1.6.1

Combine

- 2 x

$- 2 x$ and

\frac{6}{6}

$\frac{6}{6}$ .

\frac{- 2 x \cdot 6}{6} + \frac{6 x^{2} + y^{2} + 8 y + 16}{6} + 1 = 5

$\frac{- 2 x \cdot 6}{6} + \frac{6 x^{2} + y^{2} + 8 y + 16}{6} + 1 = 5$

Step 1.6.2

Combine the numerators over the common denominator.

\frac{- 2 x \cdot 6 + 6 x^{2} + y^{2} + 8 y + 16}{6} + 1 = 5

$\frac{- 2 x \cdot 6 + 6 x^{2} + y^{2} + 8 y + 16}{6} + 1 = 5$

\frac{- 2 x \cdot 6 + 6 x^{2} + y^{2} + 8 y + 16}{6} + 1 = 5

$\frac{- 2 x \cdot 6 + 6 x^{2} + y^{2} + 8 y + 16}{6} + 1 = 5$

Step 1.7

Multiply

6

$6$ by

- 2

$- 2$ .

\frac{- 12 x + 6 x^{2} + y^{2} + 8 y + 16}{6} + 1 = 5

$\frac{- 12 x + 6 x^{2} + y^{2} + 8 y + 16}{6} + 1 = 5$

Step 1.8

Simplify terms.

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Step 1.8.1

Write

1

$1$ as a fraction with a common denominator.

\frac{- 12 x + 6 x^{2} + y^{2} + 8 y + 16}{6} + \frac{6}{6} = 5

$\frac{- 12 x + 6 x^{2} + y^{2} + 8 y + 16}{6} + \frac{6}{6} = 5$

Step 1.8.2

Combine the numerators over the common denominator.

\frac{- 12 x + 6 x^{2} + y^{2} + 8 y + 16 + 6}{6} = 5

$\frac{- 12 x + 6 x^{2} + y^{2} + 8 y + 16 + 6}{6} = 5$

Step 1.8.3

Add

16

$16$ and

6

$6$ .

\frac{- 12 x + 6 x^{2} + y^{2} + 8 y + 22}{6} = 5

$\frac{- 12 x + 6 x^{2} + y^{2} + 8 y + 22}{6} = 5$

Step 1.8.4

Factor

- 1

$- 1$ out of

- 12 x

$- 12 x$ .

\frac{- (12 x) + 6 x^{2} + y^{2} + 8 y + 22}{6} = 5

$\frac{- (12 x) + 6 x^{2} + y^{2} + 8 y + 22}{6} = 5$

Step 1.8.5

Factor

- 1

$- 1$ out of

6 x^{2}

$6 x^{2}$ .

\frac{- (12 x) - (- 6 x^{2}) + y^{2} + 8 y + 22}{6} = 5

$\frac{- (12 x) - (- 6 x^{2}) + y^{2} + 8 y + 22}{6} = 5$

Step 1.8.6

Factor

- 1

$- 1$ out of

- (12 x) - (- 6 x^{2})

$- (12 x) - (- 6 x^{2})$ .

\frac{- (12 x - 6 x^{2}) + y^{2} + 8 y + 22}{6} = 5

$\frac{- (12 x - 6 x^{2}) + y^{2} + 8 y + 22}{6} = 5$

Step 1.8.7

Factor

- 1

$- 1$ out of

y^{2}

$y^{2}$ .

\frac{- (12 x - 6 x^{2}) - 1 (- y^{2}) + 8 y + 22}{6} = 5

$\frac{- (12 x - 6 x^{2}) - 1 (- y^{2}) + 8 y + 22}{6} = 5$

Step 1.8.8

Factor

- 1

$- 1$ out of

- (12 x - 6 x^{2}) - 1 (- y^{2})

$- (12 x - 6 x^{2}) - 1 (- y^{2})$ .

\frac{- (12 x - 6 x^{2} - y^{2}) + 8 y + 22}{6} = 5

$\frac{- (12 x - 6 x^{2} - y^{2}) + 8 y + 22}{6} = 5$

Step 1.8.9

Factor

$- 1$ out of

$8 y$ .

$\frac{- (12 x - 6 x^{2} - y^{2}) - (- 8 y) + 22}{6} = 5$

Step 1.8.10

Factor

$- 1$ out of

$- (12 x - 6 x^{2} - y^{2}) - (- 8 y)$ .

$\frac{- (12 x - 6 x^{2} - y^{2} - 8 y) + 22}{6} = 5$

Step 1.8.11

Rewrite

$22$ as

$- 1 (- 22)$ .

$\frac{- (12 x - 6 x^{2} - y^{2} - 8 y) - 1 \cdot - 22}{6} = 5$

Step 1.8.12

Factor

$- 1$ out of

$- (12 x - 6 x^{2} - y^{2} - 8 y) - 1 (- 22)$ .

$\frac{- (12 x - 6 x^{2} - y^{2} - 8 y - 22)}{6} = 5$

Step 1.8.13

Simplify the expression.

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Step 1.8.13.1

Rewrite

$- (12 x - 6 x^{2} - y^{2} - 8 y - 22)$ as

$- 1 (12 x - 6 x^{2} - y^{2} - 8 y - 22)$ .

$\frac{- 1 (12 x - 6 x^{2} - y^{2} - 8 y - 22)}{6} = 5$

Step 1.8.13.2

Move the negative in front of the fraction.

$- \frac{12 x - 6 x^{2} - y^{2} - 8 y - 22}{6} = 5$

Step 2

Multiply both sides by

$6$ .

$- \frac{12 x - 6 x^{2} - y^{2} - 8 y - 22}{6} \cdot 6 = 5 \cdot 6$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify

$- \frac{12 x - 6 x^{2} - y^{2} - 8 y - 22}{6} \cdot 6$ .

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Step 3.1.1.1

Cancel the common factor of

$6$ .

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Step 3.1.1.1.1

Move the leading negative in

$- \frac{12 x - 6 x^{2} - y^{2} - 8 y - 22}{6}$ into the numerator.

$\frac{- (12 x - 6 x^{2} - y^{2} - 8 y - 22)}{6} \cdot 6 = 5 \cdot 6$

Step 3.1.1.1.2

Cancel the common factor.

$\frac{- (12 x - 6 x^{2} - y^{2} - 8 y - 22)}{6} \cdot 6 = 5 \cdot 6$

Step 3.1.1.1.3

Rewrite the expression.

$- (12 x - 6 x^{2} - y^{2} - 8 y - 22) = 5 \cdot 6$

Step 3.1.1.2

Apply the distributive property.

$- (12 x) - (- 6 x^{2}) - - y^{2} - (- 8 y) - - 22 = 5 \cdot 6$

Step 3.1.1.3

Simplify.

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Step 3.1.1.3.1

Multiply

$12$ by

$- 1$ .

$- 12 x - (- 6 x^{2}) - - y^{2} - (- 8 y) - - 22 = 5 \cdot 6$

Step 3.1.1.3.2

Multiply

$- 6$ by

$- 1$ .

$- 12 x + 6 x^{2} - - y^{2} - (- 8 y) - - 22 = 5 \cdot 6$

Step 3.1.1.3.3

Multiply

$- - y^{2}$ .

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Step 3.1.1.3.3.1

Multiply

$- 1$ by

$- 1$ .

$- 12 x + 6 x^{2} + 1 y^{2} - (- 8 y) - - 22 = 5 \cdot 6$

Step 3.1.1.3.3.2

Multiply

$y^{2}$ by

$1$ .

$- 12 x + 6 x^{2} + y^{2} - (- 8 y) - - 22 = 5 \cdot 6$

Step 3.1.1.3.4

Multiply

$- 8$ by

$- 1$ .

$- 12 x + 6 x^{2} + y^{2} + 8 y - - 22 = 5 \cdot 6$

Step 3.1.1.3.5

Multiply

$- 1$ by

$- 22$ .

$- 12 x + 6 x^{2} + y^{2} + 8 y + 22 = 5 \cdot 6$

Step 3.1.1.4

Move

$- 12 x$ .

$6 x^{2} + y^{2} - 12 x + 8 y + 22 = 5 \cdot 6$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply

$5$ by

$6$ .

$6 x^{2} + y^{2} - 12 x + 8 y + 22 = 30$

Step 4

Set the equation equal to zero.

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Step 4.1

Subtract

$30$ from both sides of the equation.

$6 x^{2} + y^{2} - 12 x + 8 y + 22 - 30 = 0$

Step 4.2

Subtract

$30$ from

$22$ .

$6 x^{2} + y^{2} - 12 x + 8 y - 8 = 0$

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