Precalculus Examples

$2 x^{2} + 2 y^{2} - 2 x - 2 y = 7$

Step 1

Divide both sides of the equation by $2$ .

$x^{2} + y^{2} - x - y = \frac{7}{2}$

Step 2

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

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Step 2.1

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

$a = 1$

$b = - 1$

$c = 0$

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 $d = \frac{b}{2 a}$ .

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

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 $- 1 (1)$ .

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

Step 2.3.2.1.2

Cancel the common factor.

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

Step 2.3.2.1.3

Rewrite the expression.

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

Step 2.3.2.2

Move the negative in front of the fraction.

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

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 $e = c - \frac{b^{2}}{4 a}$ .

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

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

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

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

Step 2.4.2.1.2

Multiply $4$ by $1$ .

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

Step 2.4.2.2

Subtract $\frac{1}{4}$ from $0$ .

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

Step 2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x - \frac{1}{2})}^{2} - \frac{1}{4}$ .

${(x - \frac{1}{2})}^{2} - \frac{1}{4}$

Step 3

Substitute ${(x - \frac{1}{2})}^{2} - \frac{1}{4}$ for $x^{2} - x$ in the equation $x^{2} + y^{2} - x - y = \frac{7}{2}$ .

${(x - \frac{1}{2})}^{2} - \frac{1}{4} + y^{2} - y = \frac{7}{2}$

Step 4

Move $- \frac{1}{4}$ to the right side of the equation by adding $\frac{1}{4}$ to both sides.

${(x - \frac{1}{2})}^{2} + y^{2} - y = \frac{7}{2} + \frac{1}{4}$

Step 5

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

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Step 5.1

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

$a = 1$

$b = - 1$

$c = 0$

Step 5.2

Consider the vertex form of a parabola.

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

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

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Step 5.3.2.1.1

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

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

Step 5.3.2.1.2

Cancel the common factor.

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

Step 5.3.2.1.3

Rewrite the expression.

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

Step 5.3.2.2

Move the negative in front of the fraction.

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

Step 5.4

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

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Step 5.4.1

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

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

Step 5.4.2

Simplify the right side.

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Step 5.4.2.1

Simplify each term.

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Step 5.4.2.1.1

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

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

Step 5.4.2.1.2

Multiply $4$ by $1$ .

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

Step 5.4.2.2

Subtract $\frac{1}{4}$ from $0$ .

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

Step 5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y - \frac{1}{2})}^{2} - \frac{1}{4}$ .

${(y - \frac{1}{2})}^{2} - \frac{1}{4}$

Step 6

Substitute ${(y - \frac{1}{2})}^{2} - \frac{1}{4}$ for $y^{2} - y$ in the equation $x^{2} + y^{2} - x - y = \frac{7}{2}$ .

${(x - \frac{1}{2})}^{2} + {(y - \frac{1}{2})}^{2} - \frac{1}{4} = \frac{7}{2} + \frac{1}{4}$

Step 7

Move $- \frac{1}{4}$ to the right side of the equation by adding $\frac{1}{4}$ to both sides.

${(x - \frac{1}{2})}^{2} + {(y - \frac{1}{2})}^{2} = \frac{7}{2} + \frac{1}{4} + \frac{1}{4}$

Step 8

Simplify $\frac{7}{2} + \frac{1}{4} + \frac{1}{4}$ .

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Step 8.1

Combine the numerators over the common denominator.

${(x - \frac{1}{2})}^{2} + {(y - \frac{1}{2})}^{2} = \frac{7}{2} + \frac{1 + 1}{4}$

Step 8.2

Add $1$ and $1$ .

${(x - \frac{1}{2})}^{2} + {(y - \frac{1}{2})}^{2} = \frac{7}{2} + \frac{2}{4}$

Step 8.3

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

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Step 8.3.1

Factor $2$ out of $2$ .

${(x - \frac{1}{2})}^{2} + {(y - \frac{1}{2})}^{2} = \frac{7}{2} + \frac{2 (1)}{4}$

Step 8.3.2

Cancel the common factors.

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Step 8.3.2.1

Factor $2$ out of $4$ .

${(x - \frac{1}{2})}^{2} + {(y - \frac{1}{2})}^{2} = \frac{7}{2} + \frac{2 \cdot 1}{2 \cdot 2}$

Step 8.3.2.2

Cancel the common factor.

${(x - \frac{1}{2})}^{2} + {(y - \frac{1}{2})}^{2} = \frac{7}{2} + \frac{2 \cdot 1}{2 \cdot 2}$

Step 8.3.2.3

Rewrite the expression.

${(x - \frac{1}{2})}^{2} + {(y - \frac{1}{2})}^{2} = \frac{7}{2} + \frac{1}{2}$

Step 8.4

Combine the numerators over the common denominator.

${(x - \frac{1}{2})}^{2} + {(y - \frac{1}{2})}^{2} = \frac{7 + 1}{2}$

Step 8.5

Simplify the expression.

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Step 8.5.1

Add $7$ and $1$ .

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

Step 8.5.2

Divide $8$ by $2$ .

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

Step 9

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

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

Step 10

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.

$r = 2$

$h = \frac{1}{2}$

$k = \frac{1}{2}$

Step 11

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

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

Step 12

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

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

Radius: $2$

Step 13