Examples

$x^{2} + y^{2} + 1 + 2 x - y = 0$

Step 1

Subtract $1$ from both sides of the equation.

$x^{2} + y^{2} + 2 x - y = - 1$

Step 2

Complete the square for $x^{2} + 2 x$ .

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Step 2.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 1$

$b = 2$

$c = 0$

Step 2.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 2.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 2.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{2}{2 \cdot 1}$

Step 2.3.2

Cancel the common factor of $2$ .

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Step 2.3.2.1

Cancel the common factor.

$d = \frac{2}{2 \cdot 1}$

Step 2.3.2.2

Rewrite the expression.

$d = 1$

Step 2.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 2.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{2^{2}}{4 \cdot 1}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify each term.

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Step 2.4.2.1.1

Raise $2$ to the power of $2$ .

$e = 0 - \frac{4}{4 \cdot 1}$

Step 2.4.2.1.2

Multiply $4$ by $1$ .

$e = 0 - \frac{4}{4}$

Step 2.4.2.1.3

Cancel the common factor of $4$ .

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Step 2.4.2.1.3.1

Cancel the common factor.

$e = 0 - \frac{4}{4}$

Step 2.4.2.1.3.2

Rewrite the expression.

$e = 0 - 1 \cdot 1$

Step 2.4.2.1.4

Multiply $- 1$ by $1$ .

$e = 0 - 1$

Step 2.4.2.2

Subtract $1$ from $0$ .

$e = - 1$

Step 2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + 1)}^{2} - 1$ .

${(x + 1)}^{2} - 1$

Step 3

Substitute ${(x + 1)}^{2} - 1$ for $x^{2} + 2 x$ in the equation $x^{2} + y^{2} + 2 x - y = - 1$ .

${(x + 1)}^{2} - 1 + y^{2} - y = - 1$

Step 4

Move $- 1$ to the right side of the equation by adding $1$ to both sides.

${(x + 1)}^{2} + y^{2} - y = - 1 + 1$

Step 5

Complete the square for $y^{2} - y$ .

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Step 5.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 1$

$b = - 1$

$c = 0$

Step 5.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 5.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 5.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 1}{2 \cdot 1}$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Cancel the common factor of $- 1$ and $1$ .

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Step 5.3.2.1.1

Rewrite $- 1$ as $- 1 (1)$ .

$d = \frac{- 1 (1)}{2 \cdot 1}$

Step 5.3.2.1.2

Cancel the common factor.

$d = \frac{- 1 \cdot 1}{2 \cdot 1}$

Step 5.3.2.1.3

Rewrite the expression.

$d = \frac{- 1}{2}$

Step 5.3.2.2

Move the negative in front of the fraction.

$d = - \frac{1}{2}$

Step 5.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 5.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 1)}^{2}}{4 \cdot 1}$

Step 5.4.2

Simplify the right side.

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Step 5.4.2.1

Simplify each term.

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Step 5.4.2.1.1

Raise $- 1$ to the power of $2$ .

$e = 0 - \frac{1}{4 \cdot 1}$

Step 5.4.2.1.2

Multiply $4$ by $1$ .

$e = 0 - \frac{1}{4}$

Step 5.4.2.2

Subtract $\frac{1}{4}$ from $0$ .

$e = - \frac{1}{4}$

Step 5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y - \frac{1}{2})}^{2} - \frac{1}{4}$ .

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

Step 6

Substitute ${(y - \frac{1}{2})}^{2} - \frac{1}{4}$ for $y^{2} - y$ in the equation $x^{2} + y^{2} + 2 x - y = - 1$ .

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

Step 7

Move $- \frac{1}{4}$ to the right side of the equation by adding $\frac{1}{4}$ to both sides.

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

Step 8

Simplify $- 1 + 1 + \frac{1}{4}$ .

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Step 8.1

Find the common denominator.

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Step 8.1.1

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

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

Step 8.1.2

Multiply $\frac{- 1}{1}$ by $\frac{4}{4}$ .

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

Step 8.1.3

Multiply $\frac{- 1}{1}$ by $\frac{4}{4}$ .

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

Step 8.1.4

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

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

Step 8.1.5

Multiply $\frac{1}{1}$ by $\frac{4}{4}$ .

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

Step 8.1.6

Multiply $\frac{1}{1}$ by $\frac{4}{4}$ .

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

Step 8.2

Combine the numerators over the common denominator.

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

Step 8.3

Simplify the expression.

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Step 8.3.1

Multiply $- 1$ by $4$ .

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

Step 8.3.2

Add $- 4$ and $4$ .

${(x + 1)}^{2} + {(y - \frac{1}{2})}^{2} = \frac{0 + 1}{4}$

Step 8.3.3

Add $0$ and $1$ .

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

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