Precalculus Examples

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

Step 1

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

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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

Step 1.1.2

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

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Step 1.1.2.1

Apply the distributive property.

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

Step 1.1.2.2

Apply the distributive property.

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

Step 1.1.2.3

Apply the distributive property.

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

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 $x$ .

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

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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