Examples

4 x^{2} + 4 y^{2} - 16 x - 24 y + 48 = 0

$4 x^{2} + 4 y^{2} - 16 x - 24 y + 48 = 0$

Step 1

Subtract

48

$48$ from both sides of the equation.

4 x^{2} + 4 y^{2} - 16 x - 24 y = - 48

$4 x^{2} + 4 y^{2} - 16 x - 24 y = - 48$

Step 2

Divide both sides of the equation by

4

$4$ .

x^{2} + y^{2} - 4 x - 6 y = - 12

$x^{2} + y^{2} - 4 x - 6 y = - 12$

Step 3

Complete the square for

x^{2} - 4 x

$x^{2} - 4 x$ .

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Step 3.1

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 3.2

Consider the vertex form of a parabola.

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

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

Step 3.3

Find the value of

d

$d$ using the formula

d = \frac{b}{2 a}

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

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Step 3.3.1

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 3.3.2

Cancel the common factor of

- 4

$- 4$ and

2

$2$ .

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Step 3.3.2.1

Factor

2

$2$ out of

- 4

$- 4$ .

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

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

Step 3.3.2.2

Cancel the common factors.

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Step 3.3.2.2.1

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 3.3.2.2.2

Cancel the common factor.

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

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

Step 3.3.2.2.3

Rewrite the expression.

d = \frac{- 2}{1}

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

Step 3.3.2.2.4

Divide

- 2

$- 2$ by

1

$1$ .

d = - 2

$d = - 2$

d = - 2

$d = - 2$

d = - 2

$d = - 2$

d = - 2

$d = - 2$

Step 3.4

Find the value of

e

$e$ using the formula

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

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

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Step 3.4.1

Substitute the values of

c

$c$ ,

b

$b$ and

a

$a$ into the formula

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

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

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

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

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify each term.

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Step 3.4.2.1.1

Cancel the common factor of

{(- 4)}^{2}

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

4

$4$ .

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Step 3.4.2.1.1.1

Rewrite

- 4

$- 4$ as

- 1 (4)

$- 1 (4)$ .

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

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

Step 3.4.2.1.1.2

Apply the product rule to

- 1 (4)

$- 1 (4)$ .

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

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

Step 3.4.2.1.1.3

Raise

- 1

$- 1$ to the power of

2

$2$ .

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

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

Step 3.4.2.1.1.4

Multiply

4^{2}

$4^{2}$ by

1

$1$ .

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

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

Step 3.4.2.1.1.5

Factor

4

$4$ out of

4^{2}

$4^{2}$ .

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

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

Step 3.4.2.1.1.6

Cancel the common factors.

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Step 3.4.2.1.1.6.1

Factor

4

$4$ out of

4 \cdot 1

$4 \cdot 1$ .

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

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

Step 3.4.2.1.1.6.2

Cancel the common factor.

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

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

Step 3.4.2.1.1.6.3

Rewrite the expression.

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

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

Step 3.4.2.1.1.6.4

Divide

4

$4$ by

1

$1$ .

e = 0 - 1 \cdot 4

$e = 0 - 1 \cdot 4$

e = 0 - 1 \cdot 4

$e = 0 - 1 \cdot 4$

e = 0 - 1 \cdot 4

$e = 0 - 1 \cdot 4$

Step 3.4.2.1.2

Multiply

- 1

$- 1$ by

4

$4$ .

e = 0 - 4

$e = 0 - 4$

e = 0 - 4

$e = 0 - 4$

Step 3.4.2.2

Subtract

4

$4$ from

0

$0$ .

e = - 4

$e = - 4$

e = - 4

$e = - 4$

e = - 4

$e = - 4$

Step 3.5

Substitute the values of

a

$a$ ,

d

$d$ , and

e

$e$ into the vertex form

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

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

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

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

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

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

Step 4

Substitute

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

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

x^{2} - 4 x

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

x^{2} + y^{2} - 4 x - 6 y = - 12

$x^{2} + y^{2} - 4 x - 6 y = - 12$ .

{(x - 2)}^{2} - 4 + y^{2} - 6 y = - 12

${(x - 2)}^{2} - 4 + y^{2} - 6 y = - 12$

Step 5

Move

- 4

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

4

$4$ to both sides.

{(x - 2)}^{2} + y^{2} - 6 y = - 12 + 4

${(x - 2)}^{2} + y^{2} - 6 y = - 12 + 4$

Step 6

Complete the square for

y^{2} - 6 y

$y^{2} - 6 y$ .

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Step 6.1

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 = - 6

$b = - 6$

c = 0

$c = 0$

Step 6.2

Consider the vertex form of a parabola.

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

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

Step 6.3

Find the value of

d

$d$ using the formula

d = \frac{b}{2 a}

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

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Step 6.3.1

Substitute the values of

a

$a$ and

b

$b$ into the formula

d = \frac{b}{2 a}

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

d = \frac{- 6}{2 \cdot 1}

$d = \frac{- 6}{2 \cdot 1}$

Step 6.3.2

Cancel the common factor of

- 6

$- 6$ and

2

$2$ .

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Step 6.3.2.1

Factor

2

$2$ out of

- 6

$- 6$ .

d = \frac{2 \cdot - 3}{2 \cdot 1}

$d = \frac{2 \cdot - 3}{2 \cdot 1}$

Step 6.3.2.2

Cancel the common factors.

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Step 6.3.2.2.1

Factor

2

$2$ out of

2 \cdot 1

$2 \cdot 1$ .

d = \frac{2 \cdot - 3}{2 (1)}

$d = \frac{2 \cdot - 3}{2 (1)}$

Step 6.3.2.2.2

Cancel the common factor.

d = \frac{2 \cdot - 3}{2 \cdot 1}

$d = \frac{2 \cdot - 3}{2 \cdot 1}$

Step 6.3.2.2.3

Rewrite the expression.

d = \frac{- 3}{1}

$d = \frac{- 3}{1}$

Step 6.3.2.2.4

Divide

- 3

$- 3$ by

1

$1$ .

d = - 3

$d = - 3$

d = - 3

$d = - 3$

d = - 3

$d = - 3$

d = - 3

$d = - 3$

Step 6.4

Find the value of

e

$e$ using the formula

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

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

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Step 6.4.1

Substitute the values of

c

$c$ ,

b

$b$ and

a

$a$ into the formula

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

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

e = 0 - \frac{{(- 6)}^{2}}{4 \cdot 1}

$e = 0 - \frac{{(- 6)}^{2}}{4 \cdot 1}$

Step 6.4.2

Simplify the right side.

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Step 6.4.2.1

Simplify each term.

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Step 6.4.2.1.1

Raise

- 6

$- 6$ to the power of

2

$2$ .

e = 0 - \frac{36}{4 \cdot 1}

$e = 0 - \frac{36}{4 \cdot 1}$

Step 6.4.2.1.2

Multiply

4

$4$ by

1

$1$ .

e = 0 - \frac{36}{4}

$e = 0 - \frac{36}{4}$

Step 6.4.2.1.3

Divide

36

$36$ by

4

$4$ .

e = 0 - 1 \cdot 9

$e = 0 - 1 \cdot 9$

Step 6.4.2.1.4

Multiply

- 1

$- 1$ by

9

$9$ .

e = 0 - 9

$e = 0 - 9$

e = 0 - 9

$e = 0 - 9$

Step 6.4.2.2

Subtract

9

$9$ from

0

$0$ .

e = - 9

$e = - 9$

e = - 9

$e = - 9$

e = - 9

$e = - 9$

Step 6.5

Substitute the values of

a

$a$ ,

d

$d$ , and

e

$e$ into the vertex form

{(y - 3)}^{2} - 9

${(y - 3)}^{2} - 9$ .

{(y - 3)}^{2} - 9

${(y - 3)}^{2} - 9$

{(y - 3)}^{2} - 9

${(y - 3)}^{2} - 9$

Step 7

Substitute

${(y - 3)}^{2} - 9$ for

$y^{2} - 6 y$ in the equation

$x^{2} + y^{2} - 4 x - 6 y = - 12$ .

${(x - 2)}^{2} + {(y - 3)}^{2} - 9 = - 12 + 4$

Step 8

Move

$- 9$ to the right side of the equation by adding

$9$ to both sides.

${(x - 2)}^{2} + {(y - 3)}^{2} = - 12 + 4 + 9$

Step 9

Simplify

$- 12 + 4 + 9$ .

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Step 9.1

Add

$- 12$ and

$4$ .

${(x - 2)}^{2} + {(y - 3)}^{2} = - 8 + 9$

Step 9.2

Add

$- 8$ and

$9$ .

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

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