Precalculus Examples

$9 x^{2} + 9 y^{2} - 6 x + 24 y = 19$

Step 1

Divide both sides of the equation by $9$ .

$x^{2} + y^{2} - \frac{2 x}{3} + \frac{8 y}{3} = \frac{19}{9}$

Step 2

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

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Step 2.1

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 2.2

Consider the vertex form of a parabola.

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

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

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Step 2.3.2.1.1

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

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

Step 2.3.2.1.2

Cancel the common factor.

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

Step 2.3.2.1.3

Rewrite the expression.

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

Step 2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.3.2.3

Cancel the common factor of $2$ .

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Step 2.3.2.3.1

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

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

Step 2.3.2.3.2

Factor $2$ out of $- 2$ .

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

Step 2.3.2.3.3

Cancel the common factor.

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

Step 2.3.2.3.4

Rewrite the expression.

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

Step 2.3.2.4

Move the negative in front of the fraction.

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

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

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

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

Simplify the numerator.

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Step 2.4.2.1.1.1

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

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

Step 2.4.2.1.1.2

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

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

Step 2.4.2.1.1.3

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

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

Step 2.4.2.1.1.4

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

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

Step 2.4.2.1.1.5

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

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

Step 2.4.2.1.1.6

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

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

Step 2.4.2.1.2

Multiply $4$ by $1$ .

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

Step 2.4.2.1.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.4.2.1.4

Cancel the common factor of $4$ .

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Step 2.4.2.1.4.1

Cancel the common factor.

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

Step 2.4.2.1.4.2

Rewrite the expression.

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

Step 2.4.2.2

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

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

Step 2.5

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

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

Step 3

Substitute ${(x - \frac{1}{3})}^{2} - \frac{1}{9}$ for $x^{2} - \frac{2 x}{3}$ in the equation $x^{2} + y^{2} - \frac{2 x}{3} + \frac{8 y}{3} = \frac{19}{9}$ .

${(x - \frac{1}{3})}^{2} - \frac{1}{9} + y^{2} + \frac{8 y}{3} = \frac{19}{9}$

Step 4

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

${(x - \frac{1}{3})}^{2} + y^{2} + \frac{8 y}{3} = \frac{19}{9} + \frac{1}{9}$

Step 5

Complete the square for $y^{2} + \frac{8 y}{3}$ .

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Step 5.1

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

$a = 1$

$b = \frac{8}{3}$

$c = 0$

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

$d = \frac{\frac{8}{3}}{2 \cdot 1}$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{8}{3} \cdot \frac{1}{2 \cdot 1}$

Step 5.3.2.2

Cancel the common factor of $2$ .

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Step 5.3.2.2.1

Factor $2$ out of $8$ .

$d = \frac{2 (4)}{3} \cdot \frac{1}{2 \cdot 1}$

Step 5.3.2.2.2

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

$d = \frac{2 (4)}{3} \cdot \frac{1}{2 (1)}$

Step 5.3.2.2.3

Cancel the common factor.

$d = \frac{2 \cdot 4}{3} \cdot \frac{1}{2 \cdot 1}$

Step 5.3.2.2.4

Rewrite the expression.

$d = \frac{4}{3}$

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

$e = 0 - \frac{{(\frac{8}{3})}^{2}}{4 \cdot 1}$

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

Simplify the numerator.

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Step 5.4.2.1.1.1

Apply the product rule to $\frac{8}{3}$ .

$e = 0 - \frac{\frac{8^{2}}{3^{2}}}{4 \cdot 1}$

Step 5.4.2.1.1.2

Raise $8$ to the power of $2$ .

$e = 0 - \frac{\frac{64}{3^{2}}}{4 \cdot 1}$

Step 5.4.2.1.1.3

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

$e = 0 - \frac{\frac{64}{9}}{4 \cdot 1}$

Step 5.4.2.1.2

Multiply $4$ by $1$ .

$e = 0 - \frac{\frac{64}{9}}{4}$

Step 5.4.2.1.3

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (\frac{64}{9} \cdot \frac{1}{4})$

Step 5.4.2.1.4

Cancel the common factor of $4$ .

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Step 5.4.2.1.4.1

Factor $4$ out of $64$ .

$e = 0 - (\frac{4 (16)}{9} \cdot \frac{1}{4})$

Step 5.4.2.1.4.2

Cancel the common factor.

$e = 0 - (\frac{4 \cdot 16}{9} \cdot \frac{1}{4})$

Step 5.4.2.1.4.3

Rewrite the expression.

$e = 0 - \frac{16}{9}$

Step 5.4.2.2

Subtract $\frac{16}{9}$ from $0$ .

$e = - \frac{16}{9}$

Step 5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y + \frac{4}{3})}^{2} - \frac{16}{9}$ .

${(y + \frac{4}{3})}^{2} - \frac{16}{9}$

Step 6

Substitute ${(y + \frac{4}{3})}^{2} - \frac{16}{9}$ for $y^{2} + \frac{8 y}{3}$ in the equation $x^{2} + y^{2} - \frac{2 x}{3} + \frac{8 y}{3} = \frac{19}{9}$ .

${(x - \frac{1}{3})}^{2} + {(y + \frac{4}{3})}^{2} - \frac{16}{9} = \frac{19}{9} + \frac{1}{9}$

Step 7

Move $- \frac{16}{9}$ to the right side of the equation by adding $\frac{16}{9}$ to both sides.

${(x - \frac{1}{3})}^{2} + {(y + \frac{4}{3})}^{2} = \frac{19}{9} + \frac{1}{9} + \frac{16}{9}$

Step 8

Simplify $\frac{19}{9} + \frac{1}{9} + \frac{16}{9}$ .

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Step 8.1

Combine the numerators over the common denominator.

${(x - \frac{1}{3})}^{2} + {(y + \frac{4}{3})}^{2} = \frac{19 + 1 + 16}{9}$

Step 8.2

Simplify the expression.

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Step 8.2.1

Add $19$ and $1$ .

${(x - \frac{1}{3})}^{2} + {(y + \frac{4}{3})}^{2} = \frac{20 + 16}{9}$

Step 8.2.2

Add $20$ and $16$ .

${(x - \frac{1}{3})}^{2} + {(y + \frac{4}{3})}^{2} = \frac{36}{9}$

Step 8.2.3

Divide $36$ by $9$ .

${(x - \frac{1}{3})}^{2} + {(y + \frac{4}{3})}^{2} = 4$

Step 9

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 10

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.

$r = 2$

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

$k = - \frac{4}{3}$

Step 11

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

Center: $(\frac{1}{3}, - \frac{4}{3})$

Step 12

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

Center: $(\frac{1}{3}, - \frac{4}{3})$

Radius: $2$

Step 13