Graph x^2+y^2+2x-6y-15

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Trigonometry

$x^{2} + y^{2} + 2 x - 6 y - 15$

Add $15$ to both sides of the equation.

$x^{2} + y^{2} + 2 x - 6 y = 15$

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

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

$a = 1, b = 2, c = 0$

Consider the vertex form of a parabola.

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

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Rewrite the expression.

$d = 1$

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

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Simplify each term.

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

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

Multiply $4$ by $1$ .

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

Cancel the common factor of $4$ .

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Cancel the common factor.

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

Rewrite the expression.

$e = 0 - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$e = 0 - 1$

Subtract $1$ from $0$ .

$e = - 1$

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

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

Substitute ${(x + 1)}^{2} - 1$ for $x^{2} + 2 x$ in the equation $x^{2} + y^{2} + 2 x - 6 y = 15$ .

${(x + 1)}^{2} - 1 + y^{2} - 6 y = 15$

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

${(x + 1)}^{2} + y^{2} - 6 y = 15 + 1$

Complete the square for $y^{2} - 6 y$ .

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

$a = 1, b = - 6, c = 0$

Consider the vertex form of a parabola.

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

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

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

Simplify the right side.

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Cancel the common factor of $6$ and $2$ .

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Factor $2$ out of $6$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $3$ by $1$ .

$d = - 1 \cdot 3$

Multiply $- 1$ by $3$ .

$d = - 3$

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

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Simplify each term.

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Raise $- 6$ to the power of $2$ .

$e = 0 - \frac{36}{4 (1)}$

Multiply $4$ by $1$ .

$e = 0 - \frac{36}{4}$

Divide $36$ by $4$ .

$e = 0 - 1 \cdot 9$

Multiply $- 1$ by $9$ .

$e = 0 - 9$

Subtract $9$ from $0$ .

$e = - 9$

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

${(y - 3)}^{2} - 9$

Substitute ${(y - 3)}^{2} - 9$ for $y^{2} - 6 y$ in the equation $x^{2} + y^{2} + 2 x - 6 y = 15$ .

${(x + 1)}^{2} + {(y - 3)}^{2} - 9 = 15 + 1$

Move $- 9$ to the right side of the equation by adding $9$ to both sides.

${(x + 1)}^{2} + {(y - 3)}^{2} = 15 + 1 + 9$

Simplify $15 + 1 + 9$ .

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Add $15$ and $1$ .

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

Add $16$ and $9$ .

${(x + 1)}^{2} + {(y - 3)}^{2} = 25$

This is the form of a circle. Use this form to determine the center and radius of the circle.

${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$

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.

$r = 5$

$h = - 1$

$k = 3$

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

Center: $(- 1, 3)$

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

Center: $(- 1, 3)$

Radius: $5$

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