Precalculus Examples

$4 x^{2} + 4 y^{2} - 4 x + 12 y + 1 = 0$

Step 1

Subtract $1$ from both sides of the equation.

$4 x^{2} + 4 y^{2} - 4 x + 12 y = - 1$

Step 2

Divide both sides of the equation by $4$ .

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

Step 3

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

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Step 3.1

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

$a = 1$

$b = - 1$

$c = 0$

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

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

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

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

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Step 3.3.2.1.1

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

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

Step 3.3.2.1.2

Cancel the common factor.

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

Step 3.3.2.1.3

Rewrite the expression.

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

Step 3.3.2.2

Move the negative in front of the fraction.

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

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

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

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

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

Step 3.4.2.1.2

Multiply $4$ by $1$ .

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

Step 3.4.2.2

Subtract $\frac{1}{4}$ from $0$ .

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

Step 3.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x - \frac{1}{2})}^{2} - \frac{1}{4}$ .

${(x - \frac{1}{2})}^{2} - \frac{1}{4}$

Step 4

Substitute ${(x - \frac{1}{2})}^{2} - \frac{1}{4}$ for $x^{2} - x$ in the equation $x^{2} + y^{2} - x + 3 y = - \frac{1}{4}$ .

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

Step 5

Move $- \frac{1}{4}$ to the right side of the equation by adding $\frac{1}{4}$ to both sides.

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

Step 6

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

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Step 6.1

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

$a = 1$

$b = 3$

$c = 0$

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

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

Step 6.3.2

Multiply $2$ by $1$ .

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

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

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

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

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

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

Step 6.4.2.1.2

Multiply $4$ by $1$ .

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

Step 6.4.2.2

Subtract $\frac{9}{4}$ from $0$ .

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

Step 6.5

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

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

Step 7

Substitute ${(y + \frac{3}{2})}^{2} - \frac{9}{4}$ for $y^{2} + 3 y$ in the equation $x^{2} + y^{2} - x + 3 y = - \frac{1}{4}$ .

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

Step 8

Move $- \frac{9}{4}$ to the right side of the equation by adding $\frac{9}{4}$ to both sides.

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

Step 9

Simplify $- \frac{1}{4} + \frac{1}{4} + \frac{9}{4}$ .

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Step 9.1

Combine the numerators over the common denominator.

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

Step 9.2

Simplify by adding numbers.

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Step 9.2.1

Add $- 1$ and $1$ .

${(x - \frac{1}{2})}^{2} + {(y + \frac{3}{2})}^{2} = \frac{0 + 9}{4}$

Step 9.2.2

Add $0$ and $9$ .

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

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.

$r = \frac{3}{2}$

$h = \frac{1}{2}$

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

Step 12

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

Center: $(\frac{1}{2}, - \frac{3}{2})$

Step 13

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

Center: $(\frac{1}{2}, - \frac{3}{2})$

Radius: $\frac{3}{2}$

Step 14