Precalculus Examples

$16 x^{2} + 16 y^{2} + 96 x - 40 y = 315$

Step 1

Divide both sides of the equation by $16$ .

$x^{2} + y^{2} + 6 x - \frac{5 y}{2} = \frac{315}{16}$

Step 2

Complete the square for $x^{2} + 6 x$ .

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Step 2.1

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

$a = 1$

$b = 6$

$c = 0$

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

$d = \frac{6}{2 \cdot 1}$

Step 2.3.2

Cancel the common factor of $6$ and $2$ .

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Step 2.3.2.1

Factor $2$ out of $6$ .

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

Step 2.3.2.2

Cancel the common factors.

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Step 2.3.2.2.1

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

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

Step 2.3.2.2.2

Cancel the common factor.

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

Step 2.3.2.2.3

Rewrite the expression.

$d = \frac{3}{1}$

Step 2.3.2.2.4

Divide $3$ by $1$ .

$d = 3$

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

$e = 0 - \frac{6^{2}}{4 \cdot 1}$

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 $6$ to the power of $2$ .

$e = 0 - \frac{36}{4 \cdot 1}$

Step 2.4.2.1.2

Multiply $4$ by $1$ .

$e = 0 - \frac{36}{4}$

Step 2.4.2.1.3

Divide $36$ by $4$ .

$e = 0 - 1 \cdot 9$

Step 2.4.2.1.4

Multiply $- 1$ by $9$ .

$e = 0 - 9$

Step 2.4.2.2

Subtract $9$ from $0$ .

$e = - 9$

Step 2.5

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

${(x + 3)}^{2} - 9$

Step 3

Substitute ${(x + 3)}^{2} - 9$ for $x^{2} + 6 x$ in the equation $x^{2} + y^{2} + 6 x - \frac{5 y}{2} = \frac{315}{16}$ .

${(x + 3)}^{2} - 9 + y^{2} - \frac{5 y}{2} = \frac{315}{16}$

Step 4

Move $- 9$ to the right side of the equation by adding $9$ to both sides.

${(x + 3)}^{2} + y^{2} - \frac{5 y}{2} = \frac{315}{16} + 9$

Step 5

Complete the square for $y^{2} - \frac{5 y}{2}$ .

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Step 5.1

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

$a = 1$

$b = - \frac{5}{2}$

$c = 0$

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

$d = \frac{- \frac{5}{2}}{2 \cdot 1}$

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 $- 1 (1)$ .

$d = \frac{- 1 (1) \frac{5}{2}}{2 \cdot 1}$

Step 5.3.2.1.2

Cancel the common factor.

$d = \frac{- 1 \cdot 1 \frac{5}{2}}{2 \cdot 1}$

Step 5.3.2.1.3

Rewrite the expression.

$d = \frac{- \frac{5}{2}}{2}$

Step 5.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = - \frac{5}{2} \cdot \frac{1}{2}$

Step 5.3.2.3

Multiply $- \frac{5}{2} \cdot \frac{1}{2}$ .

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Step 5.3.2.3.1

Multiply $\frac{1}{2}$ by $\frac{5}{2}$ .

$d = - \frac{5}{2 \cdot 2}$

Step 5.3.2.3.2

Multiply $2$ by $2$ .

$d = - \frac{5}{4}$

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

$e = 0 - \frac{{(- \frac{5}{2})}^{2}}{4 \cdot 1}$

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

Simplify the numerator.

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Step 5.4.2.1.1.1

Apply the product rule to $- \frac{5}{2}$ .

$e = 0 - \frac{{(- 1)}^{2} {(\frac{5}{2})}^{2}}{4 \cdot 1}$

Step 5.4.2.1.1.2

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

$e = 0 - \frac{1 {(\frac{5}{2})}^{2}}{4 \cdot 1}$

Step 5.4.2.1.1.3

Apply the product rule to $\frac{5}{2}$ .

$e = 0 - \frac{1 \frac{5^{2}}{2^{2}}}{4 \cdot 1}$

Step 5.4.2.1.1.4

Raise $5$ to the power of $2$ .

$e = 0 - \frac{1 \frac{25}{2^{2}}}{4 \cdot 1}$

Step 5.4.2.1.1.5

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

$e = 0 - \frac{1 (\frac{25}{4})}{4 \cdot 1}$

Step 5.4.2.1.1.6

Multiply $\frac{25}{4}$ by $1$ .

$e = 0 - \frac{\frac{25}{4}}{4 \cdot 1}$

Step 5.4.2.1.2

Multiply $4$ by $1$ .

$e = 0 - \frac{\frac{25}{4}}{4}$

Step 5.4.2.1.3

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (\frac{25}{4} \cdot \frac{1}{4})$

Step 5.4.2.1.4

Multiply $\frac{25}{4} \cdot \frac{1}{4}$ .

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Step 5.4.2.1.4.1

Multiply $\frac{25}{4}$ by $\frac{1}{4}$ .

$e = 0 - \frac{25}{4 \cdot 4}$

Step 5.4.2.1.4.2

Multiply $4$ by $4$ .

$e = 0 - \frac{25}{16}$

Step 5.4.2.2

Subtract $\frac{25}{16}$ from $0$ .

$e = - \frac{25}{16}$

Step 5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y - \frac{5}{4})}^{2} - \frac{25}{16}$ .

${(y - \frac{5}{4})}^{2} - \frac{25}{16}$

Step 6

Substitute ${(y - \frac{5}{4})}^{2} - \frac{25}{16}$ for $y^{2} - \frac{5 y}{2}$ in the equation $x^{2} + y^{2} + 6 x - \frac{5 y}{2} = \frac{315}{16}$ .

${(x + 3)}^{2} + {(y - \frac{5}{4})}^{2} - \frac{25}{16} = \frac{315}{16} + 9$

Step 7

Move $- \frac{25}{16}$ to the right side of the equation by adding $\frac{25}{16}$ to both sides.

${(x + 3)}^{2} + {(y - \frac{5}{4})}^{2} = \frac{315}{16} + 9 + \frac{25}{16}$

Step 8

Simplify $\frac{315}{16} + 9 + \frac{25}{16}$ .

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Step 8.1

Combine the numerators over the common denominator.

${(x + 3)}^{2} + {(y - \frac{5}{4})}^{2} = 9 + \frac{315 + 25}{16}$

Step 8.2

Add $315$ and $25$ .

${(x + 3)}^{2} + {(y - \frac{5}{4})}^{2} = 9 + \frac{340}{16}$

Step 8.3

Cancel the common factor of $340$ and $16$ .

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Step 8.3.1

Factor $4$ out of $340$ .

${(x + 3)}^{2} + {(y - \frac{5}{4})}^{2} = 9 + \frac{4 (85)}{16}$

Step 8.3.2

Cancel the common factors.

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Step 8.3.2.1

Factor $4$ out of $16$ .

${(x + 3)}^{2} + {(y - \frac{5}{4})}^{2} = 9 + \frac{4 \cdot 85}{4 \cdot 4}$

Step 8.3.2.2

Cancel the common factor.

${(x + 3)}^{2} + {(y - \frac{5}{4})}^{2} = 9 + \frac{4 \cdot 85}{4 \cdot 4}$

Step 8.3.2.3

Rewrite the expression.

${(x + 3)}^{2} + {(y - \frac{5}{4})}^{2} = 9 + \frac{85}{4}$

Step 8.4

To write $9$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

${(x + 3)}^{2} + {(y - \frac{5}{4})}^{2} = 9 \cdot \frac{4}{4} + \frac{85}{4}$

Step 8.5

Combine $9$ and $\frac{4}{4}$ .

${(x + 3)}^{2} + {(y - \frac{5}{4})}^{2} = \frac{9 \cdot 4}{4} + \frac{85}{4}$

Step 8.6

Combine the numerators over the common denominator.

${(x + 3)}^{2} + {(y - \frac{5}{4})}^{2} = \frac{9 \cdot 4 + 85}{4}$

Step 8.7

Simplify the numerator.

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Step 8.7.1

Multiply $9$ by $4$ .

${(x + 3)}^{2} + {(y - \frac{5}{4})}^{2} = \frac{36 + 85}{4}$

Step 8.7.2

Add $36$ and $85$ .

${(x + 3)}^{2} + {(y - \frac{5}{4})}^{2} = \frac{121}{4}$

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.

$r = \frac{11}{2}$

$h = - 3$

$k = \frac{5}{4}$

Step 11

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

Center: $(- 3, \frac{5}{4})$

Step 12

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

Center: $(- 3, \frac{5}{4})$

Radius: $\frac{11}{2}$

Step 13