Precalculus Examples

$x^{2} + y^{2} + \frac{1}{2} x - 2 y - 5 = 0$

Step 1

Add $5$ to both sides of the equation.

$x^{2} + y^{2} + \frac{1}{2} x - 2 y = 5$

Step 2

Complete the square for $x^{2} + \frac{1}{2} x$ .

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Step 2.1

Combine $\frac{1}{2}$ and $x$ .

$x^{2} + \frac{x}{2}$

Step 2.2

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

$a = 1$

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

$c = 0$

Step 2.3

Consider the vertex form of a parabola.

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.4.2.2

Cancel the common factor of $1$ .

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Step 2.4.2.2.1

Cancel the common factor.

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

Step 2.4.2.2.2

Rewrite the expression.

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

Step 2.4.2.3

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

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Step 2.4.2.3.1

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

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

Step 2.4.2.3.2

Multiply $2$ by $2$ .

$d = \frac{1}{4}$

Step 2.5

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

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Step 2.5.1

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

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

Step 2.5.2

Simplify the right side.

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Step 2.5.2.1

Simplify each term.

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Step 2.5.2.1.1

Simplify the numerator.

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Step 2.5.2.1.1.1

Apply the product rule to $\frac{1}{2}$ .

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

Step 2.5.2.1.1.2

One to any power is one.

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

Step 2.5.2.1.1.3

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

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

Step 2.5.2.1.2

Multiply $4$ by $1$ .

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

Step 2.5.2.1.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.5.2.1.4

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

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Step 2.5.2.1.4.1

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

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

Step 2.5.2.1.4.2

Multiply $4$ by $4$ .

$e = 0 - \frac{1}{16}$

Step 2.5.2.2

Subtract $\frac{1}{16}$ from $0$ .

$e = - \frac{1}{16}$

Step 2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + \frac{1}{4})}^{2} - \frac{1}{16}$ .

${(x + \frac{1}{4})}^{2} - \frac{1}{16}$

Step 3

Substitute ${(x + \frac{1}{4})}^{2} - \frac{1}{16}$ for $x^{2} + \frac{1}{2} x$ in the equation $x^{2} + y^{2} + \frac{1}{2} x - 2 y = 5$ .

${(x + \frac{1}{4})}^{2} - \frac{1}{16} + y^{2} - 2 y = 5$

Step 4

Move $- \frac{1}{16}$ to the right side of the equation by adding $\frac{1}{16}$ to both sides.

${(x + \frac{1}{4})}^{2} + y^{2} - 2 y = 5 + \frac{1}{16}$

Step 5

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

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Step 5.1

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

$a = 1$

$b = - 2$

$c = 0$

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

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

Step 5.3.2

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

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Step 5.3.2.1

Factor $2$ out of $- 2$ .

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

Step 5.3.2.2

Cancel the common factors.

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Step 5.3.2.2.1

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

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

Step 5.3.2.2.2

Cancel the common factor.

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

Step 5.3.2.2.3

Rewrite the expression.

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

Step 5.3.2.2.4

Divide $- 1$ by $1$ .

$d = - 1$

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

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

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

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

Step 5.4.2.1.2

Multiply $4$ by $1$ .

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

Step 5.4.2.1.3

Cancel the common factor of $4$ .

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Step 5.4.2.1.3.1

Cancel the common factor.

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

Step 5.4.2.1.3.2

Rewrite the expression.

$e = 0 - 1 \cdot 1$

Step 5.4.2.1.4

Multiply $- 1$ by $1$ .

$e = 0 - 1$

Step 5.4.2.2

Subtract $1$ from $0$ .

$e = - 1$

Step 5.5

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

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

Step 6

Substitute ${(y - 1)}^{2} - 1$ for $y^{2} - 2 y$ in the equation $x^{2} + y^{2} + \frac{1}{2} x - 2 y = 5$ .

${(x + \frac{1}{4})}^{2} + {(y - 1)}^{2} - 1 = 5 + \frac{1}{16}$

Step 7

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

${(x + \frac{1}{4})}^{2} + {(y - 1)}^{2} = 5 + \frac{1}{16} + 1$

Step 8

Simplify $5 + \frac{1}{16} + 1$ .

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Step 8.1

Find the common denominator.

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Step 8.1.1

Write $5$ as a fraction with denominator $1$ .

${(x + \frac{1}{4})}^{2} + {(y - 1)}^{2} = \frac{5}{1} + \frac{1}{16} + 1$

Step 8.1.2

Multiply $\frac{5}{1}$ by $\frac{16}{16}$ .

${(x + \frac{1}{4})}^{2} + {(y - 1)}^{2} = \frac{5}{1} \cdot \frac{16}{16} + \frac{1}{16} + 1$

Step 8.1.3

Multiply $\frac{5}{1}$ by $\frac{16}{16}$ .

${(x + \frac{1}{4})}^{2} + {(y - 1)}^{2} = \frac{5 \cdot 16}{16} + \frac{1}{16} + 1$

Step 8.1.4

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

${(x + \frac{1}{4})}^{2} + {(y - 1)}^{2} = \frac{5 \cdot 16}{16} + \frac{1}{16} + \frac{1}{1}$

Step 8.1.5

Multiply $\frac{1}{1}$ by $\frac{16}{16}$ .

${(x + \frac{1}{4})}^{2} + {(y - 1)}^{2} = \frac{5 \cdot 16}{16} + \frac{1}{16} + \frac{1}{1} \cdot \frac{16}{16}$

Step 8.1.6

Multiply $\frac{1}{1}$ by $\frac{16}{16}$ .

${(x + \frac{1}{4})}^{2} + {(y - 1)}^{2} = \frac{5 \cdot 16}{16} + \frac{1}{16} + \frac{16}{16}$

Step 8.2

Combine the numerators over the common denominator.

${(x + \frac{1}{4})}^{2} + {(y - 1)}^{2} = \frac{5 \cdot 16 + 1 + 16}{16}$

Step 8.3

Simplify the expression.

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Step 8.3.1

Multiply $5$ by $16$ .

${(x + \frac{1}{4})}^{2} + {(y - 1)}^{2} = \frac{80 + 1 + 16}{16}$

Step 8.3.2

Add $80$ and $1$ .

${(x + \frac{1}{4})}^{2} + {(y - 1)}^{2} = \frac{81 + 16}{16}$

Step 8.3.3

Add $81$ and $16$ .

${(x + \frac{1}{4})}^{2} + {(y - 1)}^{2} = \frac{97}{16}$

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.

$r = \frac{\sqrt{97}}{4}$

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

$k = 1$

Step 11

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

Center: $(- \frac{1}{4}, 1)$

Step 12

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

Center: $(- \frac{1}{4}, 1)$

Radius: $\frac{\sqrt{97}}{4}$

Step 13