Examples

x^{2} + 4 x + 2 y + y^{2} = 9

$x^{2} + 4 x + 2 y + y^{2} = 9$

Step 1

Complete the square for

x^{2} + 4 x

$x^{2} + 4 x$ .

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Step 1.1

Use the form

a x^{2} + b x + c

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

a

$a$ ,

b

$b$ , and

c

$c$ .

a = 1

$a = 1$

b = 4

$b = 4$

c = 0

$c = 0$

Step 1.2

Consider the vertex form of a parabola.

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

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

Step 1.3

Find the value of

d

$d$ using the formula

d = \frac{b}{2 a}

$d = \frac{b}{2 a}$ .

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Step 1.3.1

Substitute the values of

a

$a$ and

b

$b$ into the formula

d = \frac{b}{2 a}

$d = \frac{b}{2 a}$ .

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

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

Step 1.3.2

Cancel the common factor of

4

$4$ and

2

$2$ .

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Step 1.3.2.1

Factor

2

$2$ out of

4

$4$ .

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

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

Step 1.3.2.2

Cancel the common factors.

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Step 1.3.2.2.1

Factor

2

$2$ out of

2 \cdot 1

$2 \cdot 1$ .

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

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

Step 1.3.2.2.2

Cancel the common factor.

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

Step 1.3.2.2.3

Rewrite the expression.

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

Step 1.3.2.2.4

Divide

$2$ by

$1$ .

$d = 2$

Step 1.4

Find the value of

$e$ using the formula

$e = c - \frac{b^{2}}{4 a}$ .

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Step 1.4.1

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 1.4.2

Simplify the right side.

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Step 1.4.2.1

Simplify each term.

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Step 1.4.2.1.1

Cancel the common factor of

$4^{2}$ and

$4$ .

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Step 1.4.2.1.1.1

Factor

$4$ out of

$4^{2}$ .

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

Step 1.4.2.1.1.2

Cancel the common factors.

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Step 1.4.2.1.1.2.1

Factor

$4$ out of

$4 \cdot 1$ .

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

Step 1.4.2.1.1.2.2

Cancel the common factor.

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

Step 1.4.2.1.1.2.3

Rewrite the expression.

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

Step 1.4.2.1.1.2.4

Divide

$4$ by

$1$ .

$e = 0 - 1 \cdot 4$

Step 1.4.2.1.2

Multiply

$- 1$ by

$4$ .

$e = 0 - 4$

Step 1.4.2.2

Subtract

$4$ from

$0$ .

$e = - 4$

Step 1.5

Substitute the values of

$a$ ,

$d$ , and

$e$ into the vertex form

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

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

Step 2

Substitute

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

$x^{2} + 4 x$ in the equation

$x^{2} + 4 x + 2 y + y^{2} = 9$ .

${(x + 2)}^{2} - 4 + 2 y + y^{2} = 9$

Step 3

Move

$- 4$ to the right side of the equation by adding

$4$ to both sides.

${(x + 2)}^{2} + 2 y + y^{2} = 9 + 4$

Step 4

Complete the square for

$2 y + y^{2}$ .

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Step 4.1

Reorder

$2 y$ and

$y^{2}$ .

$y^{2} + 2 y$

Step 4.2

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 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

$d = \frac{b}{2 a}$ .

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

Step 4.4.2

Cancel the common factor of

$2$ .

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Step 4.4.2.1

Cancel the common factor.

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

Step 4.4.2.2

Rewrite the expression.

$d = 1$

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

$e = c - \frac{b^{2}}{4 a}$ .

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

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

Raise

$2$ to the power of

$2$ .

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

Step 4.5.2.1.2

Multiply

$4$ by

$1$ .

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

Step 4.5.2.1.3

Cancel the common factor of

$4$ .

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Step 4.5.2.1.3.1

Cancel the common factor.

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

Step 4.5.2.1.3.2

Rewrite the expression.

$e = 0 - 1 \cdot 1$

Step 4.5.2.1.4

Multiply

$- 1$ by

$1$ .

$e = 0 - 1$

Step 4.5.2.2

Subtract

$1$ from

$0$ .

$e = - 1$

Step 4.6

Substitute the values of

$a$ ,

$d$ , and

$e$ into the vertex form

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

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

Step 5

Substitute

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

$2 y + y^{2}$ in the equation

$x^{2} + 4 x + 2 y + y^{2} = 9$ .

${(x + 2)}^{2} + {(y + 1)}^{2} - 1 = 9 + 4$

Step 6

Move

$- 1$ to the right side of the equation by adding

$1$ to both sides.

${(x + 2)}^{2} + {(y + 1)}^{2} = 9 + 4 + 1$

Step 7

Simplify

$9 + 4 + 1$ .

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Step 7.1

Add

$9$ and

$4$ .

${(x + 2)}^{2} + {(y + 1)}^{2} = 13 + 1$

Step 7.2

Add

$13$ and

$1$ .

${(x + 2)}^{2} + {(y + 1)}^{2} = 14$

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