Calculus Examples

$x^{2} + y^{2} - 6 x = - 4 y + 68$

Step 1

Add $4 y$ to both sides of the equation.

$x^{2} + y^{2} - 6 x + 4 y = 68$

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 $x^{2} + y^{2} - 6 x + 4 y = 68$ .

${(x - 3)}^{2} - 9 + y^{2} + 4 y = 68$

Step 4

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

${(x - 3)}^{2} + y^{2} + 4 y = 68 + 9$

Step 5

Complete the square for $y^{2} + 4 y$ .

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Step 5.1

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

$a = 1$

$b = 4$

$c = 0$

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

$d = \frac{4}{2 \cdot 1}$

Step 5.3.2

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

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Step 5.3.2.1

Factor $2$ out of $4$ .

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

Step 5.3.2.2

Cancel the common factors.

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Step 5.3.2.2.1

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

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

Step 5.3.2.2.2

Cancel the common factor.

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

Step 5.3.2.2.3

Rewrite the expression.

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

Step 5.3.2.2.4

Divide $2$ by $1$ .

$d = 2$

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

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

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

Cancel the common factor of $4^{2}$ and $4$ .

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Step 5.4.2.1.1.1

Factor $4$ out of $4^{2}$ .

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

Step 5.4.2.1.1.2

Cancel the common factors.

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Step 5.4.2.1.1.2.1

Factor $4$ out of $4 \cdot 1$ .

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

Step 5.4.2.1.1.2.2

Cancel the common factor.

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

Step 5.4.2.1.1.2.3

Rewrite the expression.

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

Step 5.4.2.1.1.2.4

Divide $4$ by $1$ .

$e = 0 - 1 \cdot 4$

Step 5.4.2.1.2

Multiply $- 1$ by $4$ .

$e = 0 - 4$

Step 5.4.2.2

Subtract $4$ from $0$ .

$e = - 4$

Step 5.5

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

${(y + 2)}^{2} - 4$

Step 6

Substitute ${(y + 2)}^{2} - 4$ for $y^{2} + 4 y$ in the equation $x^{2} + y^{2} - 6 x + 4 y = 68$ .

${(x - 3)}^{2} + {(y + 2)}^{2} - 4 = 68 + 9$

Step 7

Move $- 4$ to the right side of the equation by adding $4$ to both sides.

${(x - 3)}^{2} + {(y + 2)}^{2} = 68 + 9 + 4$

Step 8

Simplify $68 + 9 + 4$ .

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Step 8.1

Add $68$ and $9$ .

${(x - 3)}^{2} + {(y + 2)}^{2} = 77 + 4$

Step 8.2

Add $77$ and $4$ .

${(x - 3)}^{2} + {(y + 2)}^{2} = 81$

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.

$r = 9$

$h = 3$

$k = - 2$

Step 11

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

Center: $(3, - 2)$

Step 12

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

Center: $(3, - 2)$

Radius: $9$

Step 13