Examples

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

Step 1

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

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Step 1.1

Reorder $2 y$ and $y^{2}$ .

$y^{2} + 2 y$

Step 1.2

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 1.3

Consider the vertex form of a parabola.

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

Cancel the common factor of $2$ .

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Step 1.4.2.1

Cancel the common factor.

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

Step 1.4.2.2

Rewrite the expression.

$d = 1$

Step 1.5

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

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Step 1.5.1

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 1.5.2

Simplify the right side.

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Step 1.5.2.1

Simplify each term.

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Step 1.5.2.1.1

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

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

Step 1.5.2.1.2

Multiply $4$ by $1$ .

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

Step 1.5.2.1.3

Cancel the common factor of $4$ .

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Step 1.5.2.1.3.1

Cancel the common factor.

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

Step 1.5.2.1.3.2

Rewrite the expression.

$e = 0 - 1 \cdot 1$

Step 1.5.2.1.4

Multiply $- 1$ by $1$ .

$e = 0 - 1$

Step 1.5.2.2

Subtract $1$ from $0$ .

$e = - 1$

Step 1.6

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

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

Step 2

Substitute ${(y + 1)}^{2} - 1$ for $2 y + y^{2}$ in the equation $2 y - 4 x + x^{2} + y^{2} = 3$ .

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

Step 3

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

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

Step 4

Complete the square for $- 4 x + x^{2}$ .

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Step 4.1

Reorder $- 4 x$ and $x^{2}$ .

$x^{2} - 4 x$

Step 4.2

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

$a = 1$

$b = - 4$

$c = 0$

Step 4.3

Consider the vertex form of a parabola.

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

Step 4.4

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

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Step 4.4.1

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

$d = \frac{- 4}{2 \cdot 1}$

Step 4.4.2

Cancel the common factor of $- 4$ and $2$ .

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Step 4.4.2.1

Factor $2$ out of $- 4$ .

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

Step 4.4.2.2

Cancel the common factors.

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Step 4.4.2.2.1

Factor $2$ out of $2 \cdot 1$ .

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

Step 4.4.2.2.2

Cancel the common factor.

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

Step 4.4.2.2.3

Rewrite the expression.

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

Step 4.4.2.2.4

Divide $- 2$ by $1$ .

$d = - 2$

Step 4.5

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

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Step 4.5.1

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

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

Step 4.5.2

Simplify the right side.

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Step 4.5.2.1

Simplify each term.

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Step 4.5.2.1.1

Cancel the common factor of ${(- 4)}^{2}$ and $4$ .

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Step 4.5.2.1.1.1

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 4.5.2.1.1.2

Apply the product rule to $- 1 (4)$ .

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

Step 4.5.2.1.1.3

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

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

Step 4.5.2.1.1.4

Multiply $4^{2}$ by $1$ .

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

Step 4.5.2.1.1.5

Factor $4$ out of $4^{2}$ .

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

Step 4.5.2.1.1.6

Cancel the common factors.

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Step 4.5.2.1.1.6.1

Factor $4$ out of $4 \cdot 1$ .

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

Step 4.5.2.1.1.6.2

Cancel the common factor.

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

Step 4.5.2.1.1.6.3

Rewrite the expression.

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

Step 4.5.2.1.1.6.4

Divide $4$ by $1$ .

$e = 0 - 1 \cdot 4$

Step 4.5.2.1.2

Multiply $- 1$ by $4$ .

$e = 0 - 4$

Step 4.5.2.2

Subtract $4$ from $0$ .

$e = - 4$

Step 4.6

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

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

Step 5

Substitute ${(x - 2)}^{2} - 4$ for $- 4 x + x^{2}$ in the equation $2 y - 4 x + x^{2} + y^{2} = 3$ .

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

Step 6

Move $- 4$ to the right side of the equation by adding $4$ to both sides.

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

Step 7

Simplify $3 + 1 + 4$ .

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Step 7.1

Add $3$ and $1$ .

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

Step 7.2

Add $4$ and $4$ .

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

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