Precalculus Examples

$3 x^{2} + 3 y^{2} + 6 x - y = 0$

Step 1

Divide both sides of the equation by $3$ .

$x^{2} + y^{2} + 2 x - \frac{y}{3} = 0$

Step 2

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

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Step 2.1

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

$a = 1$

$b = 2$

$c = 0$

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

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

Step 2.3.2

Cancel the common factor of $2$ .

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Step 2.3.2.1

Cancel the common factor.

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

Step 2.3.2.2

Rewrite the expression.

$d = 1$

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

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

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

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

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

Step 2.4.2.1.2

Multiply $4$ by $1$ .

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

Step 2.4.2.1.3

Cancel the common factor of $4$ .

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Step 2.4.2.1.3.1

Cancel the common factor.

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

Step 2.4.2.1.3.2

Rewrite the expression.

$e = 0 - 1 \cdot 1$

Step 2.4.2.1.4

Multiply $- 1$ by $1$ .

$e = 0 - 1$

Step 2.4.2.2

Subtract $1$ from $0$ .

$e = - 1$

Step 2.5

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

${(x + 1)}^{2} - 1$

Step 3

Substitute ${(x + 1)}^{2} - 1$ for $x^{2} + 2 x$ in the equation $x^{2} + y^{2} + 2 x - \frac{y}{3} = 0$ .

${(x + 1)}^{2} - 1 + y^{2} - \frac{y}{3} = 0$

Step 4

Move $- 1$ to the right side of the equation by adding $1$ to both sides.

${(x + 1)}^{2} + y^{2} - \frac{y}{3} = 0 + 1$

Step 5

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

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Step 5.1

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

$a = 1$

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

$c = 0$

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

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

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

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

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Step 5.3.2.1.1

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

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

Step 5.3.2.1.2

Cancel the common factor.

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

Step 5.3.2.1.3

Rewrite the expression.

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

Step 5.3.2.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3.2.3

Multiply $- \frac{1}{3} \cdot \frac{1}{2}$ .

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Step 5.3.2.3.1

Multiply $\frac{1}{2}$ by $\frac{1}{3}$ .

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

Step 5.3.2.3.2

Multiply $2$ by $3$ .

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

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

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

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 $- \frac{1}{3}$ .

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

Step 5.4.2.1.1.2

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

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

Step 5.4.2.1.1.3

Apply the product rule to $\frac{1}{3}$ .

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

Step 5.4.2.1.1.4

One to any power is one.

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

Step 5.4.2.1.1.5

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

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

Step 5.4.2.1.1.6

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

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

Step 5.4.2.1.2

Multiply $4$ by $1$ .

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

Step 5.4.2.1.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.4.2.1.4

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

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Step 5.4.2.1.4.1

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

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

Step 5.4.2.1.4.2

Multiply $9$ by $4$ .

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

Step 5.4.2.2

Subtract $\frac{1}{36}$ from $0$ .

$e = - \frac{1}{36}$

Step 5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y - \frac{1}{6})}^{2} - \frac{1}{36}$ .

${(y - \frac{1}{6})}^{2} - \frac{1}{36}$

Step 6

Substitute ${(y - \frac{1}{6})}^{2} - \frac{1}{36}$ for $y^{2} - \frac{y}{3}$ in the equation $x^{2} + y^{2} + 2 x - \frac{y}{3} = 0$ .

${(x + 1)}^{2} + {(y - \frac{1}{6})}^{2} - \frac{1}{36} = 0 + 1$

Step 7

Move $- \frac{1}{36}$ to the right side of the equation by adding $\frac{1}{36}$ to both sides.

${(x + 1)}^{2} + {(y - \frac{1}{6})}^{2} = 0 + 1 + \frac{1}{36}$

Step 8

Simplify $0 + 1 + \frac{1}{36}$ .

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Step 8.1

Add $0$ and $1$ .

${(x + 1)}^{2} + {(y - \frac{1}{6})}^{2} = 1 + \frac{1}{36}$

Step 8.2

Write $1$ as a fraction with a common denominator.

${(x + 1)}^{2} + {(y - \frac{1}{6})}^{2} = \frac{36}{36} + \frac{1}{36}$

Step 8.3

Combine the numerators over the common denominator.

${(x + 1)}^{2} + {(y - \frac{1}{6})}^{2} = \frac{36 + 1}{36}$

Step 8.4

Add $36$ and $1$ .

${(x + 1)}^{2} + {(y - \frac{1}{6})}^{2} = \frac{37}{36}$

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.

$r = \frac{\sqrt{37}}{6}$

$h = - 1$

$k = \frac{1}{6}$

Step 11

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

Center: $(- 1, \frac{1}{6})$

Step 12

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

Center: $(- 1, \frac{1}{6})$

Radius: $\frac{\sqrt{37}}{6}$

Step 13