Precalculus Examples

$9 x^{2} + 54 x + 9 y^{2} - 6 y + 64 = 0$

Step 1

Subtract $64$ from both sides of the equation.

$9 x^{2} + 54 x + 9 y^{2} - 6 y = - 64$

Step 2

Divide both sides of the equation by $9$ .

$x^{2} + 6 x + y^{2} - \frac{2 y}{3} = - \frac{64}{9}$

Step 3

Complete the square for $x^{2} + 6 x$ .

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Step 3.1

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

$a = 1$

$b = 6$

$c = 0$

Step 3.2

Consider the vertex form of a parabola.

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Cancel the common factor of $6$ and $2$ .

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Step 3.3.2.1

Factor $2$ out of $6$ .

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

Step 3.3.2.2

Cancel the common factors.

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Step 3.3.2.2.1

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

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

Step 3.3.2.2.2

Cancel the common factor.

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

Step 3.3.2.2.3

Rewrite the expression.

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

Step 3.3.2.2.4

Divide $3$ by $1$ .

$d = 3$

Step 3.4

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

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Step 3.4.1

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

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

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

Raise $6$ to the power of $2$ .

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

Step 3.4.2.1.2

Multiply $4$ by $1$ .

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

Step 3.4.2.1.3

Divide $36$ by $4$ .

$e = 0 - 1 \cdot 9$

Step 3.4.2.1.4

Multiply $- 1$ by $9$ .

$e = 0 - 9$

Step 3.4.2.2

Subtract $9$ from $0$ .

$e = - 9$

Step 3.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + 3)}^{2} - 9$ .

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

Step 4

Substitute ${(x + 3)}^{2} - 9$ for $x^{2} + 6 x$ in the equation $x^{2} + 6 x + y^{2} - \frac{2 y}{3} = - \frac{64}{9}$ .

${(x + 3)}^{2} - 9 + y^{2} - \frac{2 y}{3} = - \frac{64}{9}$

Step 5

Move $- 9$ to the right side of the equation by adding $9$ to both sides.

${(x + 3)}^{2} + y^{2} - \frac{2 y}{3} = - \frac{64}{9} + 9$

Step 6

Complete the square for $y^{2} - \frac{2 y}{3}$ .

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Step 6.1

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

$a = 1$

$b = - \frac{2}{3}$

$c = 0$

Step 6.2

Consider the vertex form of a parabola.

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

Step 6.3

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

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Step 6.3.1

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

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

Step 6.3.2

Simplify the right side.

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Step 6.3.2.1

Cancel the common factor of $- 1$ and $1$ .

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Step 6.3.2.1.1

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

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

Step 6.3.2.1.2

Cancel the common factor.

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

Step 6.3.2.1.3

Rewrite the expression.

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

Step 6.3.2.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.3.2.3

Cancel the common factor of $2$ .

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Step 6.3.2.3.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

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

Step 6.3.2.3.2

Factor $2$ out of $- 2$ .

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

Step 6.3.2.3.3

Cancel the common factor.

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

Step 6.3.2.3.4

Rewrite the expression.

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

Step 6.3.2.4

Move the negative in front of the fraction.

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

Step 6.4

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

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Step 6.4.1

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

$e = 0 - \frac{{(- \frac{2}{3})}^{2}}{4 \cdot 1}$

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

Simplify the numerator.

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Step 6.4.2.1.1.1

Apply the product rule to $- \frac{2}{3}$ .

$e = 0 - \frac{{(- 1)}^{2} {(\frac{2}{3})}^{2}}{4 \cdot 1}$

Step 6.4.2.1.1.2

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

$e = 0 - \frac{1 {(\frac{2}{3})}^{2}}{4 \cdot 1}$

Step 6.4.2.1.1.3

Apply the product rule to $\frac{2}{3}$ .

$e = 0 - \frac{1 \frac{2^{2}}{3^{2}}}{4 \cdot 1}$

Step 6.4.2.1.1.4

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

$e = 0 - \frac{1 \frac{4}{3^{2}}}{4 \cdot 1}$

Step 6.4.2.1.1.5

Raise $3$ to the power of $2$ .

$e = 0 - \frac{1 (\frac{4}{9})}{4 \cdot 1}$

Step 6.4.2.1.1.6

Multiply $\frac{4}{9}$ by $1$ .

$e = 0 - \frac{\frac{4}{9}}{4 \cdot 1}$

Step 6.4.2.1.2

Multiply $4$ by $1$ .

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

Step 6.4.2.1.3

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (\frac{4}{9} \cdot \frac{1}{4})$

Step 6.4.2.1.4

Cancel the common factor of $4$ .

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Step 6.4.2.1.4.1

Cancel the common factor.

$e = 0 - (\frac{4}{9} \cdot \frac{1}{4})$

Step 6.4.2.1.4.2

Rewrite the expression.

$e = 0 - \frac{1}{9}$

Step 6.4.2.2

Subtract $\frac{1}{9}$ from $0$ .

$e = - \frac{1}{9}$

Step 6.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y - \frac{1}{3})}^{2} - \frac{1}{9}$ .

${(y - \frac{1}{3})}^{2} - \frac{1}{9}$

Step 7

Substitute ${(y - \frac{1}{3})}^{2} - \frac{1}{9}$ for $y^{2} - \frac{2 y}{3}$ in the equation $x^{2} + 6 x + y^{2} - \frac{2 y}{3} = - \frac{64}{9}$ .

${(x + 3)}^{2} + {(y - \frac{1}{3})}^{2} - \frac{1}{9} = - \frac{64}{9} + 9$

Step 8

Move $- \frac{1}{9}$ to the right side of the equation by adding $\frac{1}{9}$ to both sides.

${(x + 3)}^{2} + {(y - \frac{1}{3})}^{2} = - \frac{64}{9} + 9 + \frac{1}{9}$

Step 9

Simplify $- \frac{64}{9} + 9 + \frac{1}{9}$ .

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Step 9.1

Combine the numerators over the common denominator.

${(x + 3)}^{2} + {(y - \frac{1}{3})}^{2} = 9 + \frac{- 64 + 1}{9}$

Step 9.2

Simplify the expression.

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Step 9.2.1

Add $- 64$ and $1$ .

${(x + 3)}^{2} + {(y - \frac{1}{3})}^{2} = 9 + \frac{- 63}{9}$

Step 9.2.2

Divide $- 63$ by $9$ .

${(x + 3)}^{2} + {(y - \frac{1}{3})}^{2} = 9 - 7$

Step 9.2.3

Subtract $7$ from $9$ .

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

Step 10

This is the form of a circle. Use this form to determine the center and radius of the circle.

${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$

Step 11

Match the values in this circle to those of the standard form. The variable $r$ represents the radius of the circle, $h$ represents the x-offset from the origin, and $k$ represents the y-offset from origin.

$r = \sqrt{2}$

$h = - 3$

$k = \frac{1}{3}$

Step 12

The center of the circle is found at $(h, k)$ .

Center: $(- 3, \frac{1}{3})$

Step 13

These values represent the important values for graphing and analyzing a circle.

Center: $(- 3, \frac{1}{3})$

Radius: $\sqrt{2}$

Step 14