Graph x^2+y^2+2x-3y=8

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Trigonometry

$x^{2} + y^{2} + 2 x - 3 y = 8$

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 $x^{2} + y^{2} + 2 x - 3 y = 8$ .

${(x + 1)}^{2} - 1 + y^{2} - 3 y = 8$

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

${(x + 1)}^{2} + y^{2} - 3 y = 8 + 1$

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

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

$a = 1, b = - 3, c = 0$

Consider the vertex form of a parabola.

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

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

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

Multiply $2$ by $1$ .

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

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

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

Multiply $4$ by $1$ .

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

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

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

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

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

Substitute ${(y - \frac{3}{2})}^{2} - \frac{9}{4}$ for $y^{2} - 3 y$ in the equation $x^{2} + y^{2} + 2 x - 3 y = 8$ .

${(x + 1)}^{2} + {(y - \frac{3}{2})}^{2} - \frac{9}{4} = 8 + 1$

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

${(x + 1)}^{2} + {(y - \frac{3}{2})}^{2} = 8 + 1 + \frac{9}{4}$

Simplify $8 + 1 + \frac{9}{4}$ .

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Find the common denominator.

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Write $8$ as a fraction with denominator $1$ .

${(x + 1)}^{2} + {(y - \frac{3}{2})}^{2} = \frac{8}{1} + 1 + \frac{9}{4}$

Multiply $\frac{8}{1}$ by $\frac{4}{4}$ .

${(x + 1)}^{2} + {(y - \frac{3}{2})}^{2} = \frac{8}{1} \cdot \frac{4}{4} + 1 + \frac{9}{4}$

Multiply $\frac{8}{1}$ and $\frac{4}{4}$ .

${(x + 1)}^{2} + {(y - \frac{3}{2})}^{2} = \frac{8 \cdot 4}{4} + 1 + \frac{9}{4}$

Write $1$ as a fraction with denominator $1$ .

${(x + 1)}^{2} + {(y - \frac{3}{2})}^{2} = \frac{8 \cdot 4}{4} + \frac{1}{1} + \frac{9}{4}$

Multiply $\frac{1}{1}$ by $\frac{4}{4}$ .

${(x + 1)}^{2} + {(y - \frac{3}{2})}^{2} = \frac{8 \cdot 4}{4} + \frac{1}{1} \cdot \frac{4}{4} + \frac{9}{4}$

Multiply $\frac{1}{1}$ and $\frac{4}{4}$ .

${(x + 1)}^{2} + {(y - \frac{3}{2})}^{2} = \frac{8 \cdot 4}{4} + \frac{4}{4} + \frac{9}{4}$

Combine the numerators over the common denominator.

${(x + 1)}^{2} + {(y - \frac{3}{2})}^{2} = \frac{8 \cdot 4 + 4 + 9}{4}$

Simplify the expression.

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Multiply $8$ by $4$ .

${(x + 1)}^{2} + {(y - \frac{3}{2})}^{2} = \frac{32 + 4 + 9}{4}$

Add $32$ and $4$ .

${(x + 1)}^{2} + {(y - \frac{3}{2})}^{2} = \frac{36 + 9}{4}$

Add $36$ and $9$ .

${(x + 1)}^{2} + {(y - \frac{3}{2})}^{2} = \frac{45}{4}$

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.

$r = \frac{3 \sqrt{5}}{2}$

$h = - 1$

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

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

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

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

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

Radius: $\frac{3 \sqrt{5}}{2}$

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