Examples

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

Step 1

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

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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

Step 1.1.2

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

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Step 1.1.2.1

Apply the distributive property.

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

Step 1.1.2.2

Apply the distributive property.

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

Step 1.1.2.3

Apply the distributive property.

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

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 $x$ by $x$ .

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

Step 1.1.3.1.2

Move $- 1$ to the left of $x$ .

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

Step 1.1.3.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 1.1.3.1.4

Rewrite $- 1 x$ as $- x$ .

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

Step 1.1.3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 1.1.3.2

Subtract $x$ from $- x$ .

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

Step 1.2

To write $x^{2}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

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

Step 1.3

Simplify terms.

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Step 1.3.1

Combine $x^{2}$ and $\frac{6}{6}$ .

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

Step 1.3.2

Combine the numerators over the common denominator.

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

Step 1.4

Simplify the numerator.

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Step 1.4.1

Move $6$ to the left of $x^{2}$ .

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

Step 1.4.2

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

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

Step 1.4.3

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

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Step 1.4.3.1

Apply the distributive property.

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

Step 1.4.3.2

Apply the distributive property.

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

Step 1.4.3.3

Apply the distributive property.

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

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 $y$ by $y$ .

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

Step 1.4.4.1.2

Move $4$ to the left of $y$ .

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

Step 1.4.4.1.3

Multiply $4$ by $4$ .

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

Step 1.4.4.2

Add $4 y$ and $4 y$ .

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

Step 1.5

To write $- 2 x$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

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

Step 1.6

Simplify terms.

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Step 1.6.1

Combine $- 2 x$ and $\frac{6}{6}$ .

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

Step 1.6.2

Combine the numerators over the common denominator.

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

Step 1.7

Multiply $6$ by $- 2$ .

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

Step 1.8

Simplify terms.

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Step 1.8.1

Write $1$ as a fraction with a common denominator.

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

Step 1.8.2

Combine the numerators over the common denominator.

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

Step 1.8.3

Add $16$ and $6$ .

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

Step 1.8.4

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

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

Step 1.8.5

Factor $- 1$ out of $6 x^{2}$ .

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

Step 1.8.6

Factor $- 1$ out of $- (12 x) - (- 6 x^{2})$ .

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

Step 1.8.7

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

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

Step 1.8.8

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

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

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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