Algebra Examples

\frac{{(x - 2)}^{2}}{2} + \frac{{(y + 1)}^{2}}{4} = 8

$\frac{{(x - 2)}^{2}}{2} + \frac{{(y + 1)}^{2}}{4} = 8$

Step 1

Simplify the left side

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

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

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Step 1.1

To write

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

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

\frac{2}{2}

$\frac{2}{2}$ .

\frac{{(x - 2)}^{2}}{2} \cdot \frac{2}{2} + \frac{{(y + 1)}^{2}}{4} = 8

$\frac{{(x - 2)}^{2}}{2} \cdot \frac{2}{2} + \frac{{(y + 1)}^{2}}{4} = 8$

Step 1.2

Write each expression with a common denominator of

4

$4$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 1.2.1

Multiply

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

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

\frac{2}{2}

$\frac{2}{2}$ .

\frac{{(x - 2)}^{2} \cdot 2}{2 \cdot 2} + \frac{{(y + 1)}^{2}}{4} = 8

$\frac{{(x - 2)}^{2} \cdot 2}{2 \cdot 2} + \frac{{(y + 1)}^{2}}{4} = 8$

Step 1.2.2

Multiply

2

$2$ by

2

$2$ .

\frac{{(x - 2)}^{2} \cdot 2}{4} + \frac{{(y + 1)}^{2}}{4} = 8

$\frac{{(x - 2)}^{2} \cdot 2}{4} + \frac{{(y + 1)}^{2}}{4} = 8$

\frac{{(x - 2)}^{2} \cdot 2}{4} + \frac{{(y + 1)}^{2}}{4} = 8

$\frac{{(x - 2)}^{2} \cdot 2}{4} + \frac{{(y + 1)}^{2}}{4} = 8$

Step 1.3

Combine the numerators over the common denominator.

\frac{{(x - 2)}^{2} \cdot 2 + {(y + 1)}^{2}}{4} = 8

$\frac{{(x - 2)}^{2} \cdot 2 + {(y + 1)}^{2}}{4} = 8$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Rewrite

{(x - 2)}^{2}

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

(x - 2) (x - 2)

$(x - 2) (x - 2)$ .

\frac{(x - 2) (x - 2) \cdot 2 + {(y + 1)}^{2}}{4} = 8

$\frac{(x - 2) (x - 2) \cdot 2 + {(y + 1)}^{2}}{4} = 8$

Step 1.4.2

Expand

(x - 2) (x - 2)

$(x - 2) (x - 2)$ using the FOIL Method.

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Step 1.4.2.1

Apply the distributive property.

\frac{(x (x - 2) - 2 (x - 2)) \cdot 2 + {(y + 1)}^{2}}{4} = 8

$\frac{(x (x - 2) - 2 (x - 2)) \cdot 2 + {(y + 1)}^{2}}{4} = 8$

Step 1.4.2.2

Apply the distributive property.

\frac{(x \cdot x + x \cdot - 2 - 2 (x - 2)) \cdot 2 + {(y + 1)}^{2}}{4} = 8

$\frac{(x \cdot x + x \cdot - 2 - 2 (x - 2)) \cdot 2 + {(y + 1)}^{2}}{4} = 8$

Step 1.4.2.3

Apply the distributive property.

\frac{(x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2) \cdot 2 + {(y + 1)}^{2}}{4} = 8

$\frac{(x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2) \cdot 2 + {(y + 1)}^{2}}{4} = 8$

\frac{(x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2) \cdot 2 + {(y + 1)}^{2}}{4} = 8

$\frac{(x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2) \cdot 2 + {(y + 1)}^{2}}{4} = 8$

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 - 2 - 2 x - 2 \cdot - 2) \cdot 2 + {(y + 1)}^{2}}{4} = 8

$\frac{(x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2) \cdot 2 + {(y + 1)}^{2}}{4} = 8$

Step 1.4.3.1.2

Move

- 2

$- 2$ to the left of

x

$x$ .

\frac{(x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2) \cdot 2 + {(y + 1)}^{2}}{4} = 8

$\frac{(x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2) \cdot 2 + {(y + 1)}^{2}}{4} = 8$

Step 1.4.3.1.3

Multiply

- 2

$- 2$ by

- 2

$- 2$ .

\frac{(x^{2} - 2 x - 2 x + 4) \cdot 2 + {(y + 1)}^{2}}{4} = 8

$\frac{(x^{2} - 2 x - 2 x + 4) \cdot 2 + {(y + 1)}^{2}}{4} = 8$

\frac{(x^{2} - 2 x - 2 x + 4) \cdot 2 + {(y + 1)}^{2}}{4} = 8

$\frac{(x^{2} - 2 x - 2 x + 4) \cdot 2 + {(y + 1)}^{2}}{4} = 8$

Step 1.4.3.2

Subtract

2 x

$2 x$ from

- 2 x

$- 2 x$ .

\frac{(x^{2} - 4 x + 4) \cdot 2 + {(y + 1)}^{2}}{4} = 8

$\frac{(x^{2} - 4 x + 4) \cdot 2 + {(y + 1)}^{2}}{4} = 8$

\frac{(x^{2} - 4 x + 4) \cdot 2 + {(y + 1)}^{2}}{4} = 8

$\frac{(x^{2} - 4 x + 4) \cdot 2 + {(y + 1)}^{2}}{4} = 8$

Step 1.4.4

Apply the distributive property.

\frac{x^{2} \cdot 2 - 4 x \cdot 2 + 4 \cdot 2 + {(y + 1)}^{2}}{4} = 8

$\frac{x^{2} \cdot 2 - 4 x \cdot 2 + 4 \cdot 2 + {(y + 1)}^{2}}{4} = 8$

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} - 4 x \cdot 2 + 4 \cdot 2 + {(y + 1)}^{2}}{4} = 8

$\frac{2 \cdot x^{2} - 4 x \cdot 2 + 4 \cdot 2 + {(y + 1)}^{2}}{4} = 8$

Step 1.4.5.2

Multiply

2

$2$ by

- 4

$- 4$ .

\frac{2 \cdot x^{2} - 8 x + 4 \cdot 2 + {(y + 1)}^{2}}{4} = 8

$\frac{2 \cdot x^{2} - 8 x + 4 \cdot 2 + {(y + 1)}^{2}}{4} = 8$

Step 1.4.5.3

Multiply

4

$4$ by

2

$2$ .

\frac{2 \cdot x^{2} - 8 x + 8 + {(y + 1)}^{2}}{4} = 8

$\frac{2 \cdot x^{2} - 8 x + 8 + {(y + 1)}^{2}}{4} = 8$

\frac{2 \cdot x^{2} - 8 x + 8 + {(y + 1)}^{2}}{4} = 8

$\frac{2 \cdot x^{2} - 8 x + 8 + {(y + 1)}^{2}}{4} = 8$

Step 1.4.6

Rewrite

{(y + 1)}^{2}

${(y + 1)}^{2}$ as

(y + 1) (y + 1)

$(y + 1) (y + 1)$ .

\frac{2 x^{2} - 8 x + 8 + (y + 1) (y + 1)}{4} = 8

$\frac{2 x^{2} - 8 x + 8 + (y + 1) (y + 1)}{4} = 8$

Step 1.4.7

Expand

(y + 1) (y + 1)

$(y + 1) (y + 1)$ using the FOIL Method.

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Step 1.4.7.1

Apply the distributive property.

\frac{2 x^{2} - 8 x + 8 + y (y + 1) + 1 (y + 1)}{4} = 8

$\frac{2 x^{2} - 8 x + 8 + y (y + 1) + 1 (y + 1)}{4} = 8$

Step 1.4.7.2

Apply the distributive property.

\frac{2 x^{2} - 8 x + 8 + y \cdot y + y \cdot 1 + 1 (y + 1)}{4} = 8

$\frac{2 x^{2} - 8 x + 8 + y \cdot y + y \cdot 1 + 1 (y + 1)}{4} = 8$

Step 1.4.7.3

Apply the distributive property.

\frac{2 x^{2} - 8 x + 8 + y \cdot y + y \cdot 1 + 1 y + 1 \cdot 1}{4} = 8

$\frac{2 x^{2} - 8 x + 8 + y \cdot y + y \cdot 1 + 1 y + 1 \cdot 1}{4} = 8$

\frac{2 x^{2} - 8 x + 8 + y \cdot y + y \cdot 1 + 1 y + 1 \cdot 1}{4} = 8

$\frac{2 x^{2} - 8 x + 8 + y \cdot y + y \cdot 1 + 1 y + 1 \cdot 1}{4} = 8$

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} - 8 x + 8 + y^{2} + y \cdot 1 + 1 y + 1 \cdot 1}{4} = 8

$\frac{2 x^{2} - 8 x + 8 + y^{2} + y \cdot 1 + 1 y + 1 \cdot 1}{4} = 8$

Step 1.4.8.1.2

Multiply

y

$y$ by

1

$1$ .

\frac{2 x^{2} - 8 x + 8 + y^{2} + y + 1 y + 1 \cdot 1}{4} = 8

$\frac{2 x^{2} - 8 x + 8 + y^{2} + y + 1 y + 1 \cdot 1}{4} = 8$

Step 1.4.8.1.3

Multiply

y

$y$ by

1

$1$ .

\frac{2 x^{2} - 8 x + 8 + y^{2} + y + y + 1 \cdot 1}{4} = 8

$\frac{2 x^{2} - 8 x + 8 + y^{2} + y + y + 1 \cdot 1}{4} = 8$

Step 1.4.8.1.4

Multiply

1

$1$ by

1

$1$ .

\frac{2 x^{2} - 8 x + 8 + y^{2} + y + y + 1}{4} = 8

$\frac{2 x^{2} - 8 x + 8 + y^{2} + y + y + 1}{4} = 8$

\frac{2 x^{2} - 8 x + 8 + y^{2} + y + y + 1}{4} = 8

$\frac{2 x^{2} - 8 x + 8 + y^{2} + y + y + 1}{4} = 8$

Step 1.4.8.2

Add

y

$y$ and

y

$y$ .

\frac{2 x^{2} - 8 x + 8 + y^{2} + 2 y + 1}{4} = 8

$\frac{2 x^{2} - 8 x + 8 + y^{2} + 2 y + 1}{4} = 8$

\frac{2 x^{2} - 8 x + 8 + y^{2} + 2 y + 1}{4} = 8

$\frac{2 x^{2} - 8 x + 8 + y^{2} + 2 y + 1}{4} = 8$

Step 1.4.9

Add

8

$8$ and

1

$1$ .

\frac{2 x^{2} - 8 x + y^{2} + 2 y + 9}{4} = 8

$\frac{2 x^{2} - 8 x + y^{2} + 2 y + 9}{4} = 8$

\frac{2 x^{2} - 8 x + y^{2} + 2 y + 9}{4} = 8

$\frac{2 x^{2} - 8 x + y^{2} + 2 y + 9}{4} = 8$

\frac{2 x^{2} - 8 x + y^{2} + 2 y + 9}{4} = 8

$\frac{2 x^{2} - 8 x + y^{2} + 2 y + 9}{4} = 8$

Step 2

Multiply both sides by

4

$4$ .

\frac{2 x^{2} - 8 x + y^{2} + 2 y + 9}{4} \cdot 4 = 8 \cdot 4

$\frac{2 x^{2} - 8 x + y^{2} + 2 y + 9}{4} \cdot 4 = 8 \cdot 4$

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} - 8 x + y^{2} + 2 y + 9}{4} \cdot 4

$\frac{2 x^{2} - 8 x + y^{2} + 2 y + 9}{4} \cdot 4$ .

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Step 3.1.1.1

Cancel the common factor of

4

$4$ .

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Step 3.1.1.1.1

Cancel the common factor.

\frac{2 x^{2} - 8 x + y^{2} + 2 y + 9}{4} \cdot 4 = 8 \cdot 4

$\frac{2 x^{2} - 8 x + y^{2} + 2 y + 9}{4} \cdot 4 = 8 \cdot 4$

Step 3.1.1.1.2

Rewrite the expression.

2 x^{2} - 8 x + y^{2} + 2 y + 9 = 8 \cdot 4

$2 x^{2} - 8 x + y^{2} + 2 y + 9 = 8 \cdot 4$

2 x^{2} - 8 x + y^{2} + 2 y + 9 = 8 \cdot 4

$2 x^{2} - 8 x + y^{2} + 2 y + 9 = 8 \cdot 4$

Step 3.1.1.2

Move

- 8 x

$- 8 x$ .

2 x^{2} + y^{2} - 8 x + 2 y + 9 = 8 \cdot 4

$2 x^{2} + y^{2} - 8 x + 2 y + 9 = 8 \cdot 4$

2 x^{2} + y^{2} - 8 x + 2 y + 9 = 8 \cdot 4

$2 x^{2} + y^{2} - 8 x + 2 y + 9 = 8 \cdot 4$

2 x^{2} + y^{2} - 8 x + 2 y + 9 = 8 \cdot 4

$2 x^{2} + y^{2} - 8 x + 2 y + 9 = 8 \cdot 4$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply

8

$8$ by

4

$4$ .

2 x^{2} + y^{2} - 8 x + 2 y + 9 = 32

$2 x^{2} + y^{2} - 8 x + 2 y + 9 = 32$

2 x^{2} + y^{2} - 8 x + 2 y + 9 = 32

$2 x^{2} + y^{2} - 8 x + 2 y + 9 = 32$

2 x^{2} + y^{2} - 8 x + 2 y + 9 = 32

$2 x^{2} + y^{2} - 8 x + 2 y + 9 = 32$

Step 4

Set the equation equal to zero.

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Step 4.1

Subtract

32

$32$ from both sides of the equation.

2 x^{2} + y^{2} - 8 x + 2 y + 9 - 32 = 0

$2 x^{2} + y^{2} - 8 x + 2 y + 9 - 32 = 0$

Step 4.2

Subtract

32

$32$ from

9

$9$ .

2 x^{2} + y^{2} - 8 x + 2 y - 23 = 0

$2 x^{2} + y^{2} - 8 x + 2 y - 23 = 0$

2 x^{2} + y^{2} - 8 x + 2 y - 23 = 0

$2 x^{2} + y^{2} - 8 x + 2 y - 23 = 0$

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