Precalculus Examples

$2 x^{2} + 2 y^{2} - 3 x = 0$

Step 1

Divide both sides of the equation by $2$ .

$x^{2} + y^{2} - \frac{3 x}{2} = 0$

Step 2

Complete the square for $x^{2} - \frac{3 x}{2}$ .

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Step 2.1

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

$a = 1$

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

$c = 0$

Step 2.2

Consider the vertex form of a parabola.

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Cancel the common factor of $- 1$ and $1$ .

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Step 2.3.2.1.1

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

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

Step 2.3.2.1.2

Cancel the common factor.

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

Step 2.3.2.1.3

Rewrite the expression.

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

Step 2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.3.2.3

Multiply $- \frac{3}{2} \cdot \frac{1}{2}$ .

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Step 2.3.2.3.1

Multiply $\frac{1}{2}$ by $\frac{3}{2}$ .

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

Step 2.3.2.3.2

Multiply $2$ by $2$ .

$d = - \frac{3}{4}$

Step 2.4

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

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Step 2.4.1

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

$e = 0 - \frac{{(- \frac{3}{2})}^{2}}{4 \cdot 1}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify each term.

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Step 2.4.2.1.1

Simplify the numerator.

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Step 2.4.2.1.1.1

Apply the product rule to $- \frac{3}{2}$ .

$e = 0 - \frac{{(- 1)}^{2} {(\frac{3}{2})}^{2}}{4 \cdot 1}$

Step 2.4.2.1.1.2

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

$e = 0 - \frac{1 {(\frac{3}{2})}^{2}}{4 \cdot 1}$

Step 2.4.2.1.1.3

Apply the product rule to $\frac{3}{2}$ .

$e = 0 - \frac{1 \frac{3^{2}}{2^{2}}}{4 \cdot 1}$

Step 2.4.2.1.1.4

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

$e = 0 - \frac{1 \frac{9}{2^{2}}}{4 \cdot 1}$

Step 2.4.2.1.1.5

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

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

Step 2.4.2.1.1.6

Multiply $\frac{9}{4}$ by $1$ .

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

Step 2.4.2.1.2

Multiply $4$ by $1$ .

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

Step 2.4.2.1.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.4.2.1.4

Multiply $\frac{9}{4} \cdot \frac{1}{4}$ .

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Step 2.4.2.1.4.1

Multiply $\frac{9}{4}$ by $\frac{1}{4}$ .

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

Step 2.4.2.1.4.2

Multiply $4$ by $4$ .

$e = 0 - \frac{9}{16}$

Step 2.4.2.2

Subtract $\frac{9}{16}$ from $0$ .

$e = - \frac{9}{16}$

Step 2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x - \frac{3}{4})}^{2} - \frac{9}{16}$ .

${(x - \frac{3}{4})}^{2} - \frac{9}{16}$

Step 3

Substitute ${(x - \frac{3}{4})}^{2} - \frac{9}{16}$ for $x^{2} - \frac{3 x}{2}$ in the equation $x^{2} + y^{2} - \frac{3 x}{2} = 0$ .

${(x - \frac{3}{4})}^{2} - \frac{9}{16} + y^{2} = 0$

Step 4

Move $- \frac{9}{16}$ to the right side of the equation by adding $\frac{9}{16}$ to both sides.

${(x - \frac{3}{4})}^{2} + y^{2} = 0 + \frac{9}{16}$

Step 5

Add $0$ and $\frac{9}{16}$ .

${(x - \frac{3}{4})}^{2} + y^{2} = \frac{9}{16}$

Step 6

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 7

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.

$r = \frac{3}{4}$

$h = \frac{3}{4}$

$k = 0$

Step 8

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

Center: $(\frac{3}{4}, 0)$

Step 9

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

Center: $(\frac{3}{4}, 0)$

Radius: $\frac{3}{4}$

Step 10