Trigonometry Examples

$3 x^{2} = 9 - 3 y^{2} - 6 y$

Step 1

Move all terms containing variables to the left side of the equation.

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Step 1.1

Add $3 y^{2}$ to both sides of the equation.

$3 x^{2} + 3 y^{2} = 9 - 6 y$

Step 1.2

Add $6 y$ to both sides of the equation.

$3 x^{2} + 3 y^{2} + 6 y = 9$

Step 2

Divide both sides of the equation by $3$ .

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

Step 3

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

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Step 3.1

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

$a = 1$

$b = 2$

$c = 0$

Step 3.2

Consider the vertex form of a parabola.

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Cancel the common factor of $2$ .

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Step 3.3.2.1

Cancel the common factor.

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

Step 3.3.2.2

Rewrite the expression.

$d = 1$

Step 3.4

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

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Step 3.4.1

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

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

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify each term.

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Step 3.4.2.1.1

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

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

Step 3.4.2.1.2

Multiply $4$ by $1$ .

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

Step 3.4.2.1.3

Cancel the common factor of $4$ .

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Step 3.4.2.1.3.1

Cancel the common factor.

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

Step 3.4.2.1.3.2

Rewrite the expression.

$e = 0 - 1 \cdot 1$

Step 3.4.2.1.4

Multiply $- 1$ by $1$ .

$e = 0 - 1$

Step 3.4.2.2

Subtract $1$ from $0$ .

$e = - 1$

Step 3.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y + 1)}^{2} - 1$ .

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

Step 4

Substitute ${(y + 1)}^{2} - 1$ for $y^{2} + 2 y$ in the equation $x^{2} + y^{2} + 2 y = 3$ .

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

Step 5

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

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

Step 6

Add $3$ and $1$ .

$x^{2} + {(y + 1)}^{2} = 4$

Step 7

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 8

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.

$r = 2$

$h = 0$

$k = - 1$

Step 9

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

Center: $(0, - 1)$

Step 10

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

Center: $(0, - 1)$

Radius: $2$

Step 11