Find the Center and Radius x^2+y^2+12x+4y+15=0

$x^{2} + y^{2} + 12 x + 4 y + 15 = 0$

Step 1

Subtract $15$ from both sides of the equation.

$x^{2} + y^{2} + 12 x + 4 y = - 15$

Step 2

Complete the square for $x^{2} + 12 x$ .

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

$a = 1$

$b = 12$

$c = 0$

Consider the vertex form of a parabola.

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

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

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

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

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

Factor $2$ out of $12$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $6$ by $1$ .

$d = 6$

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

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

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

Simplify the right side.

Simplify each term.

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

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

Multiply $4$ by $1$ .

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

Divide $144$ by $4$ .

$e = 0 - 1 \cdot 36$

Multiply $- 1$ by $36$ .

$e = 0 - 36$

Subtract $36$ from $0$ .

$e = - 36$

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

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

Step 3

Substitute ${(x + 6)}^{2} - 36$ for $x^{2} + 12 x$ in the equation $x^{2} + y^{2} + 12 x + 4 y = - 15$ .

${(x + 6)}^{2} - 36 + y^{2} + 4 y = - 15$

Step 4

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

${(x + 6)}^{2} + y^{2} + 4 y = - 15 + 36$

Step 5

Complete the square for $y^{2} + 4 y$ .

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

$a = 1$

$b = 4$

$c = 0$

Consider the vertex form of a parabola.

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

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

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

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

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2$ by $1$ .

$d = 2$

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

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

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

Simplify the right side.

Simplify each term.

Cancel the common factor of $4^{2}$ and $4$ .

Factor $4$ out of $4^{2}$ .

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

Cancel the common factors.

Factor $4$ out of $4 \cdot 1$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $4$ by $1$ .

$e = 0 - 1 \cdot 4$

Multiply $- 1$ by $4$ .

$e = 0 - 4$

Subtract $4$ from $0$ .

$e = - 4$

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

${(y + 2)}^{2} - 4$

Step 6

Substitute ${(y + 2)}^{2} - 4$ for $y^{2} + 4 y$ in the equation $x^{2} + y^{2} + 12 x + 4 y = - 15$ .

${(x + 6)}^{2} + {(y + 2)}^{2} - 4 = - 15 + 36$

Step 7

Move $- 4$ to the right side of the equation by adding $4$ to both sides.

${(x + 6)}^{2} + {(y + 2)}^{2} = - 15 + 36 + 4$

Step 8

Simplify $- 15 + 36 + 4$ .

Add $- 15$ and $36$ .

${(x + 6)}^{2} + {(y + 2)}^{2} = 21 + 4$

Add $21$ and $4$ .

${(x + 6)}^{2} + {(y + 2)}^{2} = 25$

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.

$r = 5$

$h = - 6$

$k = - 2$

Step 11

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

Center: $(- 6, - 2)$

Step 12

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

Center: $(- 6, - 2)$

Radius: $5$

Step 13

Find the Center and Radius x^2+y^2+12x+4y+15=0

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