Precalculus Examples

$x^{2} + y^{2} - 12 y + 20 = 0$

Step 1

Subtract $20$ from both sides of the equation.

$x^{2} + y^{2} - 12 y = - 20$

Step 2

Complete the square for $y^{2} - 12 y$ .

Tap for more steps...

Step 2.1

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

$a = 1$

$b = - 12$

$c = 0$

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

Tap for more steps...

Step 2.3.1

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

$d = \frac{- 12}{2 \cdot 1}$

Step 2.3.2

Cancel the common factor of $- 12$ and $2$ .

Tap for more steps...

Step 2.3.2.1

Factor $2$ out of $- 12$ .

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

Step 2.3.2.2

Cancel the common factors.

Tap for more steps...

Step 2.3.2.2.1

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

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

Step 2.3.2.2.2

Cancel the common factor.

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

Step 2.3.2.2.3

Rewrite the expression.

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

Step 2.3.2.2.4

Divide $- 6$ by $1$ .

$d = - 6$

Step 2.4

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

Tap for more steps...

Step 2.4.1

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

$e = 0 - \frac{{(- 12)}^{2}}{4 \cdot 1}$

Step 2.4.2

Simplify the right side.

Tap for more steps...

Step 2.4.2.1

Simplify each term.

Tap for more steps...

Step 2.4.2.1.1

Raise $- 12$ to the power of $2$ .

$e = 0 - \frac{144}{4 \cdot 1}$

Step 2.4.2.1.2

Multiply $4$ by $1$ .

$e = 0 - \frac{144}{4}$

Step 2.4.2.1.3

Divide $144$ by $4$ .

$e = 0 - 1 \cdot 36$

Step 2.4.2.1.4

Multiply $- 1$ by $36$ .

$e = 0 - 36$

Step 2.4.2.2

Subtract $36$ from $0$ .

$e = - 36$

Step 2.5

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

${(y - 6)}^{2} - 36$

Step 3

Substitute ${(y - 6)}^{2} - 36$ for $y^{2} - 12 y$ in the equation $x^{2} + y^{2} - 12 y = - 20$ .

$x^{2} + {(y - 6)}^{2} - 36 = - 20$

Step 4

Move $- 36$ to the right side of the equation by adding $36$ to both sides.

$x^{2} + {(y - 6)}^{2} = - 20 + 36$

Step 5

Add $- 20$ and $36$ .

$x^{2} + {(y - 6)}^{2} = 16$

Step 6

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 7

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.

$r = 4$

$h = 0$

$k = 6$

Step 8

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

Center: $(0, 6)$

Step 9

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

Center: $(0, 6)$

Radius: $4$

Step 10