Graph x^2+y^2+18x-4y+76=0

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Trigonometry

$x^{2} + y^{2} + 18 x - 4 y + 76 = 0$

Subtract $76$ from both sides of the equation.

$x^{2} + y^{2} + 18 x - 4 y = - 76$

Complete the square for $x^{2} + 18 x$ .

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

$a = 1, b = 18, c = 0$

Consider the vertex form of a parabola.

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

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

$d = \frac{18}{2 (1)}$

Cancel the common factor of $18$ and $2$ .

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Factor $2$ out of $18$ .

$d = \frac{2 \cdot 9}{2 \cdot 1}$

Cancel the common factors.

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

$d = \frac{2 \cdot 9}{2 (1)}$

Cancel the common factor.

$d = \frac{2 \cdot 9}{2 \cdot 1}$

Rewrite the expression.

$d = \frac{9}{1}$

Divide $9$ by $1$ .

$d = 9$

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 $18$ to the power of $2$ .

$e = 0 - \frac{324}{4 (1)}$

Multiply $4$ by $1$ .

$e = 0 - \frac{324}{4}$

Divide $324$ by $4$ .

$e = 0 - 1 \cdot 81$

Multiply $- 1$ by $81$ .

$e = 0 - 81$

Subtract $81$ from $0$ .

$e = - 81$

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

${(x + 9)}^{2} - 81$

Substitute ${(x + 9)}^{2} - 81$ for $x^{2} + 18 x$ in the equation $x^{2} + y^{2} + 18 x - 4 y = - 76$ .

${(x + 9)}^{2} - 81 + y^{2} - 4 y = - 76$

Move $- 81$ to the right side of the equation by adding $81$ to both sides.

${(x + 9)}^{2} + y^{2} - 4 y = - 76 + 81$

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

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

$a = 1, b = - 4, c = 0$

Consider the vertex form of a parabola.

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

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

$d = - \frac{4}{2 (1)}$

Simplify the right side.

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

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Factor $2$ out of $4$ .

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

Cancel the common factors.

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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 $1$ .

$d = - 1 \cdot 2$

Multiply $- 1$ by $2$ .

$d = - 2$

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

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Rewrite $- 4$ as $- 1 (4)$ .

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

Apply the product rule to $- 1 (4)$ .

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

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

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

Multiply $4^{2}$ by $1$ .

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

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

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

Cancel the common factors.

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

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

Substitute ${(y - 2)}^{2} - 4$ for $y^{2} - 4 y$ in the equation $x^{2} + y^{2} + 18 x - 4 y = - 76$ .

${(x + 9)}^{2} + {(y - 2)}^{2} - 4 = - 76 + 81$

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

${(x + 9)}^{2} + {(y - 2)}^{2} = - 76 + 81 + 4$

Simplify $- 76 + 81 + 4$ .

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Add $- 76$ and $81$ .

${(x + 9)}^{2} + {(y - 2)}^{2} = 5 + 4$

Add $5$ and $4$ .

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

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

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

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.

$r = 3$

$h = - 9$

$k = 2$

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

Center: $(- 9, 2)$

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

Center: $(- 9, 2)$

Radius: $3$

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