Precalculus Examples

\frac{{(x - 4)}^{2}}{3} + \frac{{(y + 2)}^{2}}{6} = 10

$\frac{{(x - 4)}^{2}}{3} + \frac{{(y + 2)}^{2}}{6} = 10$

Step 1

Simplify the left side

\frac{{(x - 4)}^{2}}{3} + \frac{{(y + 2)}^{2}}{6}

$\frac{{(x - 4)}^{2}}{3} + \frac{{(y + 2)}^{2}}{6}$ .

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Step 1.1

To write

\frac{{(x - 4)}^{2}}{3}

$\frac{{(x - 4)}^{2}}{3}$ as a fraction with a common denominator, multiply by

\frac{2}{2}

$\frac{2}{2}$ .

\frac{{(x - 4)}^{2}}{3} \cdot \frac{2}{2} + \frac{{(y + 2)}^{2}}{6} = 10

$\frac{{(x - 4)}^{2}}{3} \cdot \frac{2}{2} + \frac{{(y + 2)}^{2}}{6} = 10$

Step 1.2

Write each expression with a common denominator of

6

$6$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 1.2.1

Multiply

\frac{{(x - 4)}^{2}}{3}

$\frac{{(x - 4)}^{2}}{3}$ by

\frac{2}{2}

$\frac{2}{2}$ .

\frac{{(x - 4)}^{2} \cdot 2}{3 \cdot 2} + \frac{{(y + 2)}^{2}}{6} = 10

$\frac{{(x - 4)}^{2} \cdot 2}{3 \cdot 2} + \frac{{(y + 2)}^{2}}{6} = 10$

Step 1.2.2

Multiply

3

$3$ by

2

$2$ .

\frac{{(x - 4)}^{2} \cdot 2}{6} + \frac{{(y + 2)}^{2}}{6} = 10

$\frac{{(x - 4)}^{2} \cdot 2}{6} + \frac{{(y + 2)}^{2}}{6} = 10$

\frac{{(x - 4)}^{2} \cdot 2}{6} + \frac{{(y + 2)}^{2}}{6} = 10

$\frac{{(x - 4)}^{2} \cdot 2}{6} + \frac{{(y + 2)}^{2}}{6} = 10$

Step 1.3

Combine the numerators over the common denominator.

\frac{{(x - 4)}^{2} \cdot 2 + {(y + 2)}^{2}}{6} = 10

$\frac{{(x - 4)}^{2} \cdot 2 + {(y + 2)}^{2}}{6} = 10$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Rewrite

{(x - 4)}^{2}

${(x - 4)}^{2}$ as

(x - 4) (x - 4)

$(x - 4) (x - 4)$ .

\frac{(x - 4) (x - 4) \cdot 2 + {(y + 2)}^{2}}{6} = 10

$\frac{(x - 4) (x - 4) \cdot 2 + {(y + 2)}^{2}}{6} = 10$

Step 1.4.2

Expand

(x - 4) (x - 4)

$(x - 4) (x - 4)$ using the FOIL Method.

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Step 1.4.2.1

Apply the distributive property.

\frac{(x (x - 4) - 4 (x - 4)) \cdot 2 + {(y + 2)}^{2}}{6} = 10

$\frac{(x (x - 4) - 4 (x - 4)) \cdot 2 + {(y + 2)}^{2}}{6} = 10$

Step 1.4.2.2

Apply the distributive property.

\frac{(x \cdot x + x \cdot - 4 - 4 (x - 4)) \cdot 2 + {(y + 2)}^{2}}{6} = 10

$\frac{(x \cdot x + x \cdot - 4 - 4 (x - 4)) \cdot 2 + {(y + 2)}^{2}}{6} = 10$

Step 1.4.2.3

Apply the distributive property.

\frac{(x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4) \cdot 2 + {(y + 2)}^{2}}{6} = 10

$\frac{(x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4) \cdot 2 + {(y + 2)}^{2}}{6} = 10$

\frac{(x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4) \cdot 2 + {(y + 2)}^{2}}{6} = 10

$\frac{(x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4) \cdot 2 + {(y + 2)}^{2}}{6} = 10$

Step 1.4.3

Simplify and combine like terms.

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Step 1.4.3.1

Simplify each term.

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Step 1.4.3.1.1

Multiply

x

$x$ by

x

$x$ .

\frac{(x^{2} + x \cdot - 4 - 4 x - 4 \cdot - 4) \cdot 2 + {(y + 2)}^{2}}{6} = 10

$\frac{(x^{2} + x \cdot - 4 - 4 x - 4 \cdot - 4) \cdot 2 + {(y + 2)}^{2}}{6} = 10$

Step 1.4.3.1.2

Move

- 4

$- 4$ to the left of

x

$x$ .

\frac{(x^{2} - 4 \cdot x - 4 x - 4 \cdot - 4) \cdot 2 + {(y + 2)}^{2}}{6} = 10

$\frac{(x^{2} - 4 \cdot x - 4 x - 4 \cdot - 4) \cdot 2 + {(y + 2)}^{2}}{6} = 10$

Step 1.4.3.1.3

Multiply

- 4

$- 4$ by

- 4

$- 4$ .

\frac{(x^{2} - 4 x - 4 x + 16) \cdot 2 + {(y + 2)}^{2}}{6} = 10

$\frac{(x^{2} - 4 x - 4 x + 16) \cdot 2 + {(y + 2)}^{2}}{6} = 10$

\frac{(x^{2} - 4 x - 4 x + 16) \cdot 2 + {(y + 2)}^{2}}{6} = 10

$\frac{(x^{2} - 4 x - 4 x + 16) \cdot 2 + {(y + 2)}^{2}}{6} = 10$

Step 1.4.3.2

Subtract

4 x

$4 x$ from

- 4 x

$- 4 x$ .

\frac{(x^{2} - 8 x + 16) \cdot 2 + {(y + 2)}^{2}}{6} = 10

$\frac{(x^{2} - 8 x + 16) \cdot 2 + {(y + 2)}^{2}}{6} = 10$

\frac{(x^{2} - 8 x + 16) \cdot 2 + {(y + 2)}^{2}}{6} = 10

$\frac{(x^{2} - 8 x + 16) \cdot 2 + {(y + 2)}^{2}}{6} = 10$

Step 1.4.4

Apply the distributive property.

\frac{x^{2} \cdot 2 - 8 x \cdot 2 + 16 \cdot 2 + {(y + 2)}^{2}}{6} = 10

$\frac{x^{2} \cdot 2 - 8 x \cdot 2 + 16 \cdot 2 + {(y + 2)}^{2}}{6} = 10$

Step 1.4.5

Simplify.

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Step 1.4.5.1

Move

2

$2$ to the left of

x^{2}

$x^{2}$ .

\frac{2 \cdot x^{2} - 8 x \cdot 2 + 16 \cdot 2 + {(y + 2)}^{2}}{6} = 10

$\frac{2 \cdot x^{2} - 8 x \cdot 2 + 16 \cdot 2 + {(y + 2)}^{2}}{6} = 10$

Step 1.4.5.2

Multiply

2

$2$ by

- 8

$- 8$ .

\frac{2 \cdot x^{2} - 16 x + 16 \cdot 2 + {(y + 2)}^{2}}{6} = 10

$\frac{2 \cdot x^{2} - 16 x + 16 \cdot 2 + {(y + 2)}^{2}}{6} = 10$

Step 1.4.5.3

Multiply

16

$16$ by

2

$2$ .

\frac{2 \cdot x^{2} - 16 x + 32 + {(y + 2)}^{2}}{6} = 10

$\frac{2 \cdot x^{2} - 16 x + 32 + {(y + 2)}^{2}}{6} = 10$

\frac{2 \cdot x^{2} - 16 x + 32 + {(y + 2)}^{2}}{6} = 10

$\frac{2 \cdot x^{2} - 16 x + 32 + {(y + 2)}^{2}}{6} = 10$

Step 1.4.6

Rewrite

{(y + 2)}^{2}

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

(y + 2) (y + 2)

$(y + 2) (y + 2)$ .

\frac{2 x^{2} - 16 x + 32 + (y + 2) (y + 2)}{6} = 10

$\frac{2 x^{2} - 16 x + 32 + (y + 2) (y + 2)}{6} = 10$

Step 1.4.7

Expand

(y + 2) (y + 2)

$(y + 2) (y + 2)$ using the FOIL Method.

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Step 1.4.7.1

Apply the distributive property.

\frac{2 x^{2} - 16 x + 32 + y (y + 2) + 2 (y + 2)}{6} = 10

$\frac{2 x^{2} - 16 x + 32 + y (y + 2) + 2 (y + 2)}{6} = 10$

Step 1.4.7.2

Apply the distributive property.

\frac{2 x^{2} - 16 x + 32 + y \cdot y + y \cdot 2 + 2 (y + 2)}{6} = 10

$\frac{2 x^{2} - 16 x + 32 + y \cdot y + y \cdot 2 + 2 (y + 2)}{6} = 10$

Step 1.4.7.3

Apply the distributive property.

\frac{2 x^{2} - 16 x + 32 + y \cdot y + y \cdot 2 + 2 y + 2 \cdot 2}{6} = 10

$\frac{2 x^{2} - 16 x + 32 + y \cdot y + y \cdot 2 + 2 y + 2 \cdot 2}{6} = 10$

\frac{2 x^{2} - 16 x + 32 + y \cdot y + y \cdot 2 + 2 y + 2 \cdot 2}{6} = 10

$\frac{2 x^{2} - 16 x + 32 + y \cdot y + y \cdot 2 + 2 y + 2 \cdot 2}{6} = 10$

Step 1.4.8

Simplify and combine like terms.

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Step 1.4.8.1

Simplify each term.

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Step 1.4.8.1.1

Multiply

y

$y$ by

y

$y$ .

\frac{2 x^{2} - 16 x + 32 + y^{2} + y \cdot 2 + 2 y + 2 \cdot 2}{6} = 10

$\frac{2 x^{2} - 16 x + 32 + y^{2} + y \cdot 2 + 2 y + 2 \cdot 2}{6} = 10$

Step 1.4.8.1.2

Move

2

$2$ to the left of

y

$y$ .

\frac{2 x^{2} - 16 x + 32 + y^{2} + 2 \cdot y + 2 y + 2 \cdot 2}{6} = 10

$\frac{2 x^{2} - 16 x + 32 + y^{2} + 2 \cdot y + 2 y + 2 \cdot 2}{6} = 10$

Step 1.4.8.1.3

Multiply

2

$2$ by

2

$2$ .

\frac{2 x^{2} - 16 x + 32 + y^{2} + 2 y + 2 y + 4}{6} = 10

$\frac{2 x^{2} - 16 x + 32 + y^{2} + 2 y + 2 y + 4}{6} = 10$

\frac{2 x^{2} - 16 x + 32 + y^{2} + 2 y + 2 y + 4}{6} = 10

$\frac{2 x^{2} - 16 x + 32 + y^{2} + 2 y + 2 y + 4}{6} = 10$

Step 1.4.8.2

Add

2 y

$2 y$ and

2 y

$2 y$ .

\frac{2 x^{2} - 16 x + 32 + y^{2} + 4 y + 4}{6} = 10

$\frac{2 x^{2} - 16 x + 32 + y^{2} + 4 y + 4}{6} = 10$

\frac{2 x^{2} - 16 x + 32 + y^{2} + 4 y + 4}{6} = 10

$\frac{2 x^{2} - 16 x + 32 + y^{2} + 4 y + 4}{6} = 10$

Step 1.4.9

Add

32

$32$ and

4

$4$ .

\frac{2 x^{2} - 16 x + y^{2} + 4 y + 36}{6} = 10

$\frac{2 x^{2} - 16 x + y^{2} + 4 y + 36}{6} = 10$

\frac{2 x^{2} - 16 x + y^{2} + 4 y + 36}{6} = 10

$\frac{2 x^{2} - 16 x + y^{2} + 4 y + 36}{6} = 10$

\frac{2 x^{2} - 16 x + y^{2} + 4 y + 36}{6} = 10

$\frac{2 x^{2} - 16 x + y^{2} + 4 y + 36}{6} = 10$

Step 2

Multiply both sides by

6

$6$ .

\frac{2 x^{2} - 16 x + y^{2} + 4 y + 36}{6} \cdot 6 = 10 \cdot 6

$\frac{2 x^{2} - 16 x + y^{2} + 4 y + 36}{6} \cdot 6 = 10 \cdot 6$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify

\frac{2 x^{2} - 16 x + y^{2} + 4 y + 36}{6} \cdot 6

$\frac{2 x^{2} - 16 x + y^{2} + 4 y + 36}{6} \cdot 6$ .

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Step 3.1.1.1

Cancel the common factor of

6

$6$ .

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Step 3.1.1.1.1

Cancel the common factor.

$\frac{2 x^{2} - 16 x + y^{2} + 4 y + 36}{6} \cdot 6 = 10 \cdot 6$

Step 3.1.1.1.2

Rewrite the expression.

$2 x^{2} - 16 x + y^{2} + 4 y + 36 = 10 \cdot 6$

Step 3.1.1.2

Move

$- 16 x$ .

$2 x^{2} + y^{2} - 16 x + 4 y + 36 = 10 \cdot 6$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply

$10$ by

$6$ .

$2 x^{2} + y^{2} - 16 x + 4 y + 36 = 60$

Step 4

Set the equation equal to zero.

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Step 4.1

Subtract

$60$ from both sides of the equation.

$2 x^{2} + y^{2} - 16 x + 4 y + 36 - 60 = 0$

Step 4.2

Subtract

$60$ from

$36$ .

$2 x^{2} + y^{2} - 16 x + 4 y - 24 = 0$

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