Trigonometry Examples

x^{2} - 6 x + y^{2} - 32 y = - 264

$x^{2} - 6 x + y^{2} - 32 y = - 264$

Step 1

Complete the square for

x^{2} - 6 x

$x^{2} - 6 x$ .

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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 = - 6

$b = - 6$

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{- 6}{2 \cdot 1}

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

Step 1.3.2

Cancel the common factor of

- 6

$- 6$ and

2

$2$ .

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Step 1.3.2.1

Factor

2

$2$ out of

- 6

$- 6$ .

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

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

Step 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 - 3}{2 (1)}

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

Step 1.3.2.2.2

Cancel the common factor.

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

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

Step 1.3.2.2.3

Rewrite the expression.

d = \frac{- 3}{1}

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

Step 1.3.2.2.4

Divide

- 3

$- 3$ by

1

$1$ .

d = - 3

$d = - 3$

d = - 3

$d = - 3$

d = - 3

$d = - 3$

d = - 3

$d = - 3$

Step 1.4

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 1.4.1

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 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

Raise

- 6

$- 6$ to the power of

2

$2$ .

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

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

Step 1.4.2.1.2

Multiply

4

$4$ by

1

$1$ .

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

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

Step 1.4.2.1.3

Divide

36

$36$ by

4

$4$ .

e = 0 - 1 \cdot 9

$e = 0 - 1 \cdot 9$

Step 1.4.2.1.4

Multiply

- 1

$- 1$ by

9

$9$ .

e = 0 - 9

$e = 0 - 9$

e = 0 - 9

$e = 0 - 9$

Step 1.4.2.2

Subtract

9

$9$ from

0

$0$ .

e = - 9

$e = - 9$

e = - 9

$e = - 9$

e = - 9

$e = - 9$

Step 1.5

Substitute the values of

a

$a$ ,

d

$d$ , and

e

$e$ into the vertex form

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

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

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

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

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

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

Step 2

Substitute

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

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

x^{2} - 6 x

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

x^{2} - 6 x + y^{2} - 32 y = - 264

$x^{2} - 6 x + y^{2} - 32 y = - 264$ .

{(x - 3)}^{2} - 9 + y^{2} - 32 y = - 264

${(x - 3)}^{2} - 9 + y^{2} - 32 y = - 264$

Step 3

Move

- 9

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

9

$9$ to both sides.

{(x - 3)}^{2} + y^{2} - 32 y = - 264 + 9

${(x - 3)}^{2} + y^{2} - 32 y = - 264 + 9$

Step 4

Complete the square for

y^{2} - 32 y

$y^{2} - 32 y$ .

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Step 4.1

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 = - 32

$b = - 32$

c = 0

$c = 0$

Step 4.2

Consider the vertex form of a parabola.

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

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

Step 4.3

Find the value of

d

$d$ using the formula

d = \frac{b}{2 a}

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

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Step 4.3.1

Substitute the values of

a

$a$ and

b

$b$ into the formula

d = \frac{b}{2 a}

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

d = \frac{- 32}{2 \cdot 1}

$d = \frac{- 32}{2 \cdot 1}$

Step 4.3.2

Cancel the common factor of

- 32

$- 32$ and

2

$2$ .

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Step 4.3.2.1

Factor

2

$2$ out of

- 32

$- 32$ .

d = \frac{2 \cdot - 16}{2 \cdot 1}

$d = \frac{2 \cdot - 16}{2 \cdot 1}$

Step 4.3.2.2

Cancel the common factors.

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Step 4.3.2.2.1

Factor

2

$2$ out of

2 \cdot 1

$2 \cdot 1$ .

d = \frac{2 \cdot - 16}{2 (1)}

$d = \frac{2 \cdot - 16}{2 (1)}$

Step 4.3.2.2.2

Cancel the common factor.

d = \frac{2 \cdot - 16}{2 \cdot 1}

$d = \frac{2 \cdot - 16}{2 \cdot 1}$

Step 4.3.2.2.3

Rewrite the expression.

d = \frac{- 16}{1}

$d = \frac{- 16}{1}$

Step 4.3.2.2.4

Divide

- 16

$- 16$ by

1

$1$ .

d = - 16

$d = - 16$

d = - 16

$d = - 16$

d = - 16

$d = - 16$

d = - 16

$d = - 16$

Step 4.4

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 4.4.1

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{{(- 32)}^{2}}{4 \cdot 1}

$e = 0 - \frac{{(- 32)}^{2}}{4 \cdot 1}$

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Simplify each term.

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Step 4.4.2.1.1

Raise

- 32

$- 32$ to the power of

2

$2$ .

e = 0 - \frac{1024}{4 \cdot 1}

$e = 0 - \frac{1024}{4 \cdot 1}$

Step 4.4.2.1.2

Multiply

4

$4$ by

1

$1$ .

e = 0 - \frac{1024}{4}

$e = 0 - \frac{1024}{4}$

Step 4.4.2.1.3

Divide

1024

$1024$ by

4

$4$ .

e = 0 - 1 \cdot 256

$e = 0 - 1 \cdot 256$

Step 4.4.2.1.4

Multiply

- 1

$- 1$ by

256

$256$ .

e = 0 - 256

$e = 0 - 256$

e = 0 - 256

$e = 0 - 256$

Step 4.4.2.2

Subtract

256

$256$ from

0

$0$ .

e = - 256

$e = - 256$

e = - 256

$e = - 256$

e = - 256

$e = - 256$

Step 4.5

Substitute the values of

a

$a$ ,

d

$d$ , and

e

$e$ into the vertex form

{(y - 16)}^{2} - 256

${(y - 16)}^{2} - 256$ .

{(y - 16)}^{2} - 256

${(y - 16)}^{2} - 256$

{(y - 16)}^{2} - 256

${(y - 16)}^{2} - 256$

Step 5

Substitute

{(y - 16)}^{2} - 256

${(y - 16)}^{2} - 256$ for

y^{2} - 32 y

$y^{2} - 32 y$ in the equation

x^{2} - 6 x + y^{2} - 32 y = - 264

$x^{2} - 6 x + y^{2} - 32 y = - 264$ .

{(x - 3)}^{2} + {(y - 16)}^{2} - 256 = - 264 + 9

${(x - 3)}^{2} + {(y - 16)}^{2} - 256 = - 264 + 9$

Step 6

Move

- 256

$- 256$ to the right side of the equation by adding

256

$256$ to both sides.

{(x - 3)}^{2} + {(y - 16)}^{2} = - 264 + 9 + 256

${(x - 3)}^{2} + {(y - 16)}^{2} = - 264 + 9 + 256$

Step 7

Simplify

- 264 + 9 + 256

$- 264 + 9 + 256$ .

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Step 7.1

Add

- 264

$- 264$ and

9

$9$ .

{(x - 3)}^{2} + {(y - 16)}^{2} = - 255 + 256

${(x - 3)}^{2} + {(y - 16)}^{2} = - 255 + 256$

Step 7.2

Add

- 255

$- 255$ and

256

$256$ .

{(x - 3)}^{2} + {(y - 16)}^{2} = 1

${(x - 3)}^{2} + {(y - 16)}^{2} = 1$

{(x - 3)}^{2} + {(y - 16)}^{2} = 1

${(x - 3)}^{2} + {(y - 16)}^{2} = 1$