Calculus Examples

$x^{2} + y^{2} + 9 = 4 x - 10 y$

Step 1

Move all terms containing variables to the left side of the equation.

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Step 1.1

Subtract $4 x$ from both sides of the equation.

$x^{2} + y^{2} + 9 - 4 x = - 10 y$

Step 1.2

Add $10 y$ to both sides of the equation.

$x^{2} + y^{2} + 9 - 4 x + 10 y = 0$

Step 1.3

Move $9$ .

$x^{2} + y^{2} - 4 x + 10 y + 9 = 0$

Step 2

Subtract $9$ from both sides of the equation.

$x^{2} + y^{2} - 4 x + 10 y = - 9$

Step 3

Complete the square for $x^{2} - 4 x$ .

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Step 3.1

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 3.2

Consider the vertex form of a parabola.

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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Step 3.3.2.1

Factor $2$ out of $- 4$ .

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

Step 3.3.2.2

Cancel the common factors.

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Step 3.3.2.2.1

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

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

Step 3.3.2.2.2

Cancel the common factor.

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

Step 3.3.2.2.3

Rewrite the expression.

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

Step 3.3.2.2.4

Divide $- 2$ by $1$ .

$d = - 2$

Step 3.4

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

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Step 3.4.1

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 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify each term.

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Step 3.4.2.1.1

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

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Step 3.4.2.1.1.1

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 3.4.2.1.1.2

Apply the product rule to $- 1 (4)$ .

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

Step 3.4.2.1.1.3

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

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

Step 3.4.2.1.1.4

Multiply $4^{2}$ by $1$ .

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

Step 3.4.2.1.1.5

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

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

Step 3.4.2.1.1.6

Cancel the common factors.

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Step 3.4.2.1.1.6.1

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

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

Step 3.4.2.1.1.6.2

Cancel the common factor.

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

Step 3.4.2.1.1.6.3

Rewrite the expression.

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

Step 3.4.2.1.1.6.4

Divide $4$ by $1$ .

$e = 0 - 1 \cdot 4$

Step 3.4.2.1.2

Multiply $- 1$ by $4$ .

$e = 0 - 4$

Step 3.4.2.2

Subtract $4$ from $0$ .

$e = - 4$

Step 3.5

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

${(x - 2)}^{2} - 4$

Step 4

Substitute ${(x - 2)}^{2} - 4$ for $x^{2} - 4 x$ in the equation $x^{2} + y^{2} - 4 x + 10 y = - 9$ .

${(x - 2)}^{2} - 4 + y^{2} + 10 y = - 9$

Step 5

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

${(x - 2)}^{2} + y^{2} + 10 y = - 9 + 4$

Step 6

Complete the square for $y^{2} + 10 y$ .

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Step 6.1

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

$a = 1$

$b = 10$

$c = 0$

Step 6.2

Consider the vertex form of a parabola.

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

Step 6.3

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

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Step 6.3.1

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

$d = \frac{10}{2 \cdot 1}$

Step 6.3.2

Cancel the common factor of $10$ and $2$ .

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Step 6.3.2.1

Factor $2$ out of $10$ .

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

Step 6.3.2.2

Cancel the common factors.

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Step 6.3.2.2.1

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

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

Step 6.3.2.2.2

Cancel the common factor.

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

Step 6.3.2.2.3

Rewrite the expression.

$d = \frac{5}{1}$

Step 6.3.2.2.4

Divide $5$ by $1$ .

$d = 5$

Step 6.4

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

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Step 6.4.1

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

$e = 0 - \frac{10^{2}}{4 \cdot 1}$

Step 6.4.2

Simplify the right side.

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Step 6.4.2.1

Simplify each term.

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Step 6.4.2.1.1

Raise $10$ to the power of $2$ .

$e = 0 - \frac{100}{4 \cdot 1}$

Step 6.4.2.1.2

Multiply $4$ by $1$ .

$e = 0 - \frac{100}{4}$

Step 6.4.2.1.3

Divide $100$ by $4$ .

$e = 0 - 1 \cdot 25$

Step 6.4.2.1.4

Multiply $- 1$ by $25$ .

$e = 0 - 25$

Step 6.4.2.2

Subtract $25$ from $0$ .

$e = - 25$

Step 6.5

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

${(y + 5)}^{2} - 25$

Step 7

Substitute ${(y + 5)}^{2} - 25$ for $y^{2} + 10 y$ in the equation $x^{2} + y^{2} - 4 x + 10 y = - 9$ .

${(x - 2)}^{2} + {(y + 5)}^{2} - 25 = - 9 + 4$

Step 8

Move $- 25$ to the right side of the equation by adding $25$ to both sides.

${(x - 2)}^{2} + {(y + 5)}^{2} = - 9 + 4 + 25$

Step 9

Simplify $- 9 + 4 + 25$ .

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Step 9.1

Add $- 9$ and $4$ .

${(x - 2)}^{2} + {(y + 5)}^{2} = - 5 + 25$

Step 9.2

Add $- 5$ and $25$ .

${(x - 2)}^{2} + {(y + 5)}^{2} = 20$

Step 10

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 11

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.

$r = 2 \sqrt{5}$

$h = 2$

$k = - 5$

Step 12

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

Center: $(2, - 5)$

Step 13

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

Center: $(2, - 5)$

Radius: $2 \sqrt{5}$

Step 14