Examples

4 x^{2} + 9 y^{2} - 16 x - 18 y - 11 = 0

$4 x^{2} + 9 y^{2} - 16 x - 18 y - 11 = 0$

Step 1

Add

11

$11$ to both sides of the equation.

4 x^{2} + 9 y^{2} - 16 x - 18 y = 11

$4 x^{2} + 9 y^{2} - 16 x - 18 y = 11$

Step 2

Complete the square for

4 x^{2} - 16 x

$4 x^{2} - 16 x$ .

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Step 2.1

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 = 4

$a = 4$

b = - 16

$b = - 16$

c = 0

$c = 0$

Step 2.2

Consider the vertex form of a parabola.

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

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

Step 2.3

Find the value of

d

$d$ using the formula

d = \frac{b}{2 a}

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

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Step 2.3.1

Substitute the values of

a

$a$ and

b

$b$ into the formula

d = \frac{b}{2 a}

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

d = \frac{- 16}{2 \cdot 4}

$d = \frac{- 16}{2 \cdot 4}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Cancel the common factor of

- 16

$- 16$ and

2

$2$ .

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Step 2.3.2.1.1

Factor

2

$2$ out of

- 16

$- 16$ .

d = \frac{2 \cdot - 8}{2 \cdot 4}

$d = \frac{2 \cdot - 8}{2 \cdot 4}$

Step 2.3.2.1.2

Cancel the common factors.

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Step 2.3.2.1.2.1

Factor

2

$2$ out of

2 \cdot 4

$2 \cdot 4$ .

d = \frac{2 \cdot - 8}{2 (4)}

$d = \frac{2 \cdot - 8}{2 (4)}$

Step 2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 8}{2 \cdot 4}$

Step 2.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 8}{4}$

Step 2.3.2.2

Cancel the common factor of

$- 8$ and

$4$ .

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Step 2.3.2.2.1

Factor

$4$ out of

$- 8$ .

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

Step 2.3.2.2.2

Cancel the common factors.

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Step 2.3.2.2.2.1

Factor

$4$ out of

$4$ .

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

Step 2.3.2.2.2.2

Cancel the common factor.

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

Step 2.3.2.2.2.3

Rewrite the expression.

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

Step 2.3.2.2.2.4

Divide

$- 2$ by

$1$ .

$d = - 2$

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

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

$e = 0 - \frac{{(- 16)}^{2}}{4 \cdot 4}$

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

$- 16$ to the power of

$2$ .

$e = 0 - \frac{256}{4 \cdot 4}$

Step 2.4.2.1.2

Multiply

$4$ by

$4$ .

$e = 0 - \frac{256}{16}$

Step 2.4.2.1.3

Divide

$256$ by

$16$ .

$e = 0 - 1 \cdot 16$

Step 2.4.2.1.4

Multiply

$- 1$ by

$16$ .

$e = 0 - 16$

Step 2.4.2.2

Subtract

$16$ from

$0$ .

$e = - 16$

Step 2.5

Substitute the values of

$a$ ,

$d$ , and

$e$ into the vertex form

$4 {(x - 2)}^{2} - 16$ .

$4 {(x - 2)}^{2} - 16$

Step 3

Substitute

$4 {(x - 2)}^{2} - 16$ for

$4 x^{2} - 16 x$ in the equation

$4 x^{2} + 9 y^{2} - 16 x - 18 y = 11$ .

$4 {(x - 2)}^{2} - 16 + 9 y^{2} - 18 y = 11$

Step 4

Move

$- 16$ to the right side of the equation by adding

$16$ to both sides.

$4 {(x - 2)}^{2} + 9 y^{2} - 18 y = 11 + 16$

Step 5

Complete the square for

$9 y^{2} - 18 y$ .

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Step 5.1

Use the form

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

$a$ ,

$b$ , and

$c$ .

$a = 9$

$b = - 18$

$c = 0$

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

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

$d = \frac{- 18}{2 \cdot 9}$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Cancel the common factor of

$- 18$ and

$2$ .

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Step 5.3.2.1.1

Factor

$2$ out of

$- 18$ .

$d = \frac{2 \cdot - 9}{2 \cdot 9}$

Step 5.3.2.1.2

Cancel the common factors.

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Step 5.3.2.1.2.1

Factor

$2$ out of

$2 \cdot 9$ .

$d = \frac{2 \cdot - 9}{2 (9)}$

Step 5.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 9}{2 \cdot 9}$

Step 5.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 9}{9}$

Step 5.3.2.2

Cancel the common factor of

$- 9$ and

$9$ .

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Step 5.3.2.2.1

Factor

$9$ out of

$- 9$ .

$d = \frac{9 \cdot - 1}{9}$

Step 5.3.2.2.2

Cancel the common factors.

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Step 5.3.2.2.2.1

Factor

$9$ out of

$9$ .

$d = \frac{9 \cdot - 1}{9 (1)}$

Step 5.3.2.2.2.2

Cancel the common factor.

$d = \frac{9 \cdot - 1}{9 \cdot 1}$

Step 5.3.2.2.2.3

Rewrite the expression.

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

Step 5.3.2.2.2.4

Divide

$- 1$ by

$1$ .

$d = - 1$

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

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

$e = 0 - \frac{{(- 18)}^{2}}{4 \cdot 9}$

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

$- 18$ to the power of

$2$ .

$e = 0 - \frac{324}{4 \cdot 9}$

Step 5.4.2.1.2

Multiply

$4$ by

$9$ .

$e = 0 - \frac{324}{36}$

Step 5.4.2.1.3

Divide

$324$ by

$36$ .

$e = 0 - 1 \cdot 9$

Step 5.4.2.1.4

Multiply

$- 1$ by

$9$ .

$e = 0 - 9$

Step 5.4.2.2

Subtract

$9$ from

$0$ .

$e = - 9$

Step 5.5

Substitute the values of

$a$ ,

$d$ , and

$e$ into the vertex form

$9 {(y - 1)}^{2} - 9$ .

$9 {(y - 1)}^{2} - 9$

Step 6

Substitute

$9 {(y - 1)}^{2} - 9$ for

$9 y^{2} - 18 y$ in the equation

$4 x^{2} + 9 y^{2} - 16 x - 18 y = 11$ .

$4 {(x - 2)}^{2} + 9 {(y - 1)}^{2} - 9 = 11 + 16$

Step 7

Move

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

$9$ to both sides.

$4 {(x - 2)}^{2} + 9 {(y - 1)}^{2} = 11 + 16 + 9$

Step 8

Simplify

$11 + 16 + 9$ .

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Step 8.1

Add

$11$ and

$16$ .

$4 {(x - 2)}^{2} + 9 {(y - 1)}^{2} = 27 + 9$

Step 8.2

Add

$27$ and

$9$ .

$4 {(x - 2)}^{2} + 9 {(y - 1)}^{2} = 36$

Step 9

Divide each term by

$36$ to make the right side equal to one.

$\frac{4 {(x - 2)}^{2}}{36} + \frac{9 {(y - 1)}^{2}}{36} = \frac{36}{36}$

Step 10

Simplify each term in the equation in order to set the right side equal to

$1$ . The standard form of an ellipse or hyperbola requires the right side of the equation be

$1$ .

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

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