Examples

{(x - 2)}^{2} + \frac{{(y + 1)}^{2}}{4} = 3

${(x - 2)}^{2} + \frac{{(y + 1)}^{2}}{4} = 3$

Step 1

Simplify the left side

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

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

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite

{(x - 2)}^{2}

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

(x - 2) (x - 2)

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

(x - 2) (x - 2) + \frac{{(y + 1)}^{2}}{4} = 3

$(x - 2) (x - 2) + \frac{{(y + 1)}^{2}}{4} = 3$

Step 1.1.2

Expand

(x - 2) (x - 2)

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

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Step 1.1.2.1

Apply the distributive property.

x (x - 2) - 2 (x - 2) + \frac{{(y + 1)}^{2}}{4} = 3

$x (x - 2) - 2 (x - 2) + \frac{{(y + 1)}^{2}}{4} = 3$

Step 1.1.2.2

Apply the distributive property.

x \cdot x + x \cdot - 2 - 2 (x - 2) + \frac{{(y + 1)}^{2}}{4} = 3

$x \cdot x + x \cdot - 2 - 2 (x - 2) + \frac{{(y + 1)}^{2}}{4} = 3$

Step 1.1.2.3

Apply the distributive property.

x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2 + \frac{{(y + 1)}^{2}}{4} = 3

$x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2 + \frac{{(y + 1)}^{2}}{4} = 3$

x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2 + \frac{{(y + 1)}^{2}}{4} = 3

$x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2 + \frac{{(y + 1)}^{2}}{4} = 3$

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 - 2 - 2 x - 2 \cdot - 2 + \frac{{(y + 1)}^{2}}{4} = 3

$x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2 + \frac{{(y + 1)}^{2}}{4} = 3$

Step 1.1.3.1.2

Move

$- 2$ to the left of

$x$ .

$x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2 + \frac{{(y + 1)}^{2}}{4} = 3$

Step 1.1.3.1.3

Multiply

$- 2$ by

$- 2$ .

$x^{2} - 2 x - 2 x + 4 + \frac{{(y + 1)}^{2}}{4} = 3$

Step 1.1.3.2

Subtract

$2 x$ from

$- 2 x$ .

$x^{2} - 4 x + 4 + \frac{{(y + 1)}^{2}}{4} = 3$

Step 1.2

To write

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

$\frac{4}{4}$ .

$- 4 x + 4 + x^{2} \cdot \frac{4}{4} + \frac{{(y + 1)}^{2}}{4} = 3$

Step 1.3

Simplify terms.

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Step 1.3.1

Combine

$x^{2}$ and

$\frac{4}{4}$ .

$- 4 x + 4 + \frac{x^{2} \cdot 4}{4} + \frac{{(y + 1)}^{2}}{4} = 3$

Step 1.3.2

Combine the numerators over the common denominator.

$- 4 x + 4 + \frac{x^{2} \cdot 4 + {(y + 1)}^{2}}{4} = 3$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Move

$4$ to the left of

$x^{2}$ .

$- 4 x + 4 + \frac{4 \cdot x^{2} + {(y + 1)}^{2}}{4} = 3$

Step 1.4.2

Rewrite

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

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

$- 4 x + 4 + \frac{4 x^{2} + (y + 1) (y + 1)}{4} = 3$

Step 1.4.3

Expand

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

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Step 1.4.3.1

Apply the distributive property.

$- 4 x + 4 + \frac{4 x^{2} + y (y + 1) + 1 (y + 1)}{4} = 3$

Step 1.4.3.2

Apply the distributive property.

$- 4 x + 4 + \frac{4 x^{2} + y \cdot y + y \cdot 1 + 1 (y + 1)}{4} = 3$

Step 1.4.3.3

Apply the distributive property.

$- 4 x + 4 + \frac{4 x^{2} + y \cdot y + y \cdot 1 + 1 y + 1 \cdot 1}{4} = 3$

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

$y$ .

$- 4 x + 4 + \frac{4 x^{2} + y^{2} + y \cdot 1 + 1 y + 1 \cdot 1}{4} = 3$

Step 1.4.4.1.2

Multiply

$y$ by

$1$ .

$- 4 x + 4 + \frac{4 x^{2} + y^{2} + y + 1 y + 1 \cdot 1}{4} = 3$

Step 1.4.4.1.3

Multiply

$y$ by

$1$ .

$- 4 x + 4 + \frac{4 x^{2} + y^{2} + y + y + 1 \cdot 1}{4} = 3$

Step 1.4.4.1.4

Multiply

$1$ by

$1$ .

$- 4 x + 4 + \frac{4 x^{2} + y^{2} + y + y + 1}{4} = 3$

Step 1.4.4.2

Add

$y$ and

$y$ .

$- 4 x + 4 + \frac{4 x^{2} + y^{2} + 2 y + 1}{4} = 3$

Step 1.5

To write

$- 4 x$ as a fraction with a common denominator, multiply by

$\frac{4}{4}$ .

$- 4 x \cdot \frac{4}{4} + \frac{4 x^{2} + y^{2} + 2 y + 1}{4} + 4 = 3$

Step 1.6

Simplify terms.

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Step 1.6.1

Combine

$- 4 x$ and

$\frac{4}{4}$ .

$\frac{- 4 x \cdot 4}{4} + \frac{4 x^{2} + y^{2} + 2 y + 1}{4} + 4 = 3$

Step 1.6.2

Combine the numerators over the common denominator.

$\frac{- 4 x \cdot 4 + 4 x^{2} + y^{2} + 2 y + 1}{4} + 4 = 3$

Step 1.7

Multiply

$4$ by

$- 4$ .

$\frac{- 16 x + 4 x^{2} + y^{2} + 2 y + 1}{4} + 4 = 3$

Step 1.8

To write

$4$ as a fraction with a common denominator, multiply by

$\frac{4}{4}$ .

$\frac{- 16 x + 4 x^{2} + y^{2} + 2 y + 1}{4} + 4 \cdot \frac{4}{4} = 3$

Step 1.9

Simplify terms.

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Step 1.9.1

Combine

$4$ and

$\frac{4}{4}$ .

$\frac{- 16 x + 4 x^{2} + y^{2} + 2 y + 1}{4} + \frac{4 \cdot 4}{4} = 3$

Step 1.9.2

Combine the numerators over the common denominator.

$\frac{- 16 x + 4 x^{2} + y^{2} + 2 y + 1 + 4 \cdot 4}{4} = 3$

Step 1.10

Simplify the numerator.

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Step 1.10.1

Multiply

$4$ by

$4$ .

$\frac{- 16 x + 4 x^{2} + y^{2} + 2 y + 1 + 16}{4} = 3$

Step 1.10.2

Add

$1$ and

$16$ .

$\frac{- 16 x + 4 x^{2} + y^{2} + 2 y + 17}{4} = 3$

Step 1.11

Simplify with factoring out.

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Step 1.11.1

Factor

$- 1$ out of

$- 16 x$ .

$\frac{- (16 x) + 4 x^{2} + y^{2} + 2 y + 17}{4} = 3$

Step 1.11.2

Factor

$- 1$ out of

$4 x^{2}$ .

$\frac{- (16 x) - (- 4 x^{2}) + y^{2} + 2 y + 17}{4} = 3$

Step 1.11.3

Factor

$- 1$ out of

$- (16 x) - (- 4 x^{2})$ .

$\frac{- (16 x - 4 x^{2}) + y^{2} + 2 y + 17}{4} = 3$

Step 1.11.4

Factor

$- 1$ out of

$y^{2}$ .

$\frac{- (16 x - 4 x^{2}) - 1 (- y^{2}) + 2 y + 17}{4} = 3$

Step 1.11.5

Factor

$- 1$ out of

$- (16 x - 4 x^{2}) - 1 (- y^{2})$ .

$\frac{- (16 x - 4 x^{2} - y^{2}) + 2 y + 17}{4} = 3$

Step 1.11.6

Factor

$- 1$ out of

$2 y$ .

$\frac{- (16 x - 4 x^{2} - y^{2}) - (- 2 y) + 17}{4} = 3$

Step 1.11.7

Factor

$- 1$ out of

$- (16 x - 4 x^{2} - y^{2}) - (- 2 y)$ .

$\frac{- (16 x - 4 x^{2} - y^{2} - 2 y) + 17}{4} = 3$

Step 1.11.8

Rewrite

$17$ as

$- 1 (- 17)$ .

$\frac{- (16 x - 4 x^{2} - y^{2} - 2 y) - 1 \cdot - 17}{4} = 3$

Step 1.11.9

Factor

$- 1$ out of

$- (16 x - 4 x^{2} - y^{2} - 2 y) - 1 (- 17)$ .

$\frac{- (16 x - 4 x^{2} - y^{2} - 2 y - 17)}{4} = 3$

Step 1.11.10

Simplify the expression.

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Step 1.11.10.1

Rewrite

$- (16 x - 4 x^{2} - y^{2} - 2 y - 17)$ as

$- 1 (16 x - 4 x^{2} - y^{2} - 2 y - 17)$ .

$\frac{- 1 (16 x - 4 x^{2} - y^{2} - 2 y - 17)}{4} = 3$

Step 1.11.10.2

Move the negative in front of the fraction.

$- \frac{16 x - 4 x^{2} - y^{2} - 2 y - 17}{4} = 3$

Step 2

Multiply both sides by

$4$ .

$- \frac{16 x - 4 x^{2} - y^{2} - 2 y - 17}{4} \cdot 4 = 3 \cdot 4$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify

$- \frac{16 x - 4 x^{2} - y^{2} - 2 y - 17}{4} \cdot 4$ .

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Step 3.1.1.1

Cancel the common factor of

$4$ .

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Step 3.1.1.1.1

Move the leading negative in

$- \frac{16 x - 4 x^{2} - y^{2} - 2 y - 17}{4}$ into the numerator.

$\frac{- (16 x - 4 x^{2} - y^{2} - 2 y - 17)}{4} \cdot 4 = 3 \cdot 4$

Step 3.1.1.1.2

Cancel the common factor.

$\frac{- (16 x - 4 x^{2} - y^{2} - 2 y - 17)}{4} \cdot 4 = 3 \cdot 4$

Step 3.1.1.1.3

Rewrite the expression.

$- (16 x - 4 x^{2} - y^{2} - 2 y - 17) = 3 \cdot 4$

Step 3.1.1.2

Apply the distributive property.

$- (16 x) - (- 4 x^{2}) - - y^{2} - (- 2 y) - - 17 = 3 \cdot 4$

Step 3.1.1.3

Simplify.

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Step 3.1.1.3.1

Multiply

$16$ by

$- 1$ .

$- 16 x - (- 4 x^{2}) - - y^{2} - (- 2 y) - - 17 = 3 \cdot 4$

Step 3.1.1.3.2

Multiply

$- 4$ by

$- 1$ .

$- 16 x + 4 x^{2} - - y^{2} - (- 2 y) - - 17 = 3 \cdot 4$

Step 3.1.1.3.3

Multiply

$- - y^{2}$ .

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Step 3.1.1.3.3.1

Multiply

$- 1$ by

$- 1$ .

$- 16 x + 4 x^{2} + 1 y^{2} - (- 2 y) - - 17 = 3 \cdot 4$

Step 3.1.1.3.3.2

Multiply

$y^{2}$ by

$1$ .

$- 16 x + 4 x^{2} + y^{2} - (- 2 y) - - 17 = 3 \cdot 4$

Step 3.1.1.3.4

Multiply

$- 2$ by

$- 1$ .

$- 16 x + 4 x^{2} + y^{2} + 2 y - - 17 = 3 \cdot 4$

Step 3.1.1.3.5

Multiply

$- 1$ by

$- 17$ .

$- 16 x + 4 x^{2} + y^{2} + 2 y + 17 = 3 \cdot 4$

Step 3.1.1.4

Move

$- 16 x$ .

$4 x^{2} + y^{2} - 16 x + 2 y + 17 = 3 \cdot 4$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply

$3$ by

$4$ .

$4 x^{2} + y^{2} - 16 x + 2 y + 17 = 12$

Step 4

Set the equation equal to zero.

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Step 4.1

Subtract

$12$ from both sides of the equation.

$4 x^{2} + y^{2} - 16 x + 2 y + 17 - 12 = 0$

Step 4.2

Subtract

$12$ from

$17$ .

$4 x^{2} + y^{2} - 16 x + 2 y + 5 = 0$

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