Calculus Examples

$- 4 x^{2} - y^{2} + 6 x + 2 y - y + 16 - 10 x - 27 + 3 y + 5 - 3 y^{2} + 5 x^{2} - y^{2} = 9 - 6 y^{2} - 4 y - 30 + 10 x + 17 + y - 3 y - 60$

Step 1

Since $y$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$9 - 6 y^{2} - 4 y - 30 + 10 x + 17 + y - 3 y - 60 = x^{2} - 5 y^{2} - 4 x + 4 y - 6$

Step 2

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

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Step 2.1

Subtract $x^{2}$ from both sides of the equation.

$9 - 6 y^{2} - 4 y - 30 + 10 x + 17 + y - 3 y - 60 - x^{2} = - 5 y^{2} - 4 x + 4 y - 6$

Step 2.2

Add $5 y^{2}$ to both sides of the equation.

$9 - 6 y^{2} - 4 y - 30 + 10 x + 17 + y - 3 y - 60 - x^{2} + 5 y^{2} = - 4 x + 4 y - 6$

Step 2.3

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

$9 - 6 y^{2} - 4 y - 30 + 10 x + 17 + y - 3 y - 60 - x^{2} + 5 y^{2} + 4 x = 4 y - 6$

Step 2.4

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

$9 - 6 y^{2} - 4 y - 30 + 10 x + 17 + y - 3 y - 60 - x^{2} + 5 y^{2} + 4 x - 4 y = - 6$

Step 2.5

Subtract $30$ from $9$ .

$- 6 y^{2} - 4 y - 21 + 10 x + 17 + y - 3 y - 60 - x^{2} + 5 y^{2} + 4 x - 4 y = - 6$

Step 2.6

Add $- 6 y^{2}$ and $5 y^{2}$ .

$- y^{2} - 4 y - 21 + 10 x + 17 + y - 3 y - 60 - x^{2} + 4 x - 4 y = - 6$

Step 2.7

Add $- 4 y$ and $y$ .

$- y^{2} - 3 y - 21 + 10 x + 17 - 3 y - 60 - x^{2} + 4 x - 4 y = - 6$

Step 2.8

Subtract $3 y$ from $- 3 y$ .

$- y^{2} - 6 y - 21 + 10 x + 17 - 60 - x^{2} + 4 x - 4 y = - 6$

Step 2.9

Subtract $4 y$ from $- 6 y$ .

$- y^{2} - 10 y - 21 + 10 x + 17 - 60 - x^{2} + 4 x = - 6$

Step 2.10

Add $- 21$ and $17$ .

$- y^{2} - 10 y + 10 x - 4 - 60 - x^{2} + 4 x = - 6$

Step 2.11

Add $10 x$ and $4 x$ .

$- y^{2} - 10 y + 14 x - 4 - 60 - x^{2} = - 6$

Step 2.12

Subtract $60$ from $- 4$ .

$- y^{2} - 10 y + 14 x - 64 - x^{2} = - 6$

Step 2.13

Move $- 64$ .

$- y^{2} - 10 y + 14 x - x^{2} - 64 = - 6$

Step 2.14

Move $- 10 y$ .

$- y^{2} + 14 x - x^{2} - 10 y - 64 = - 6$

Step 2.15

Move $14 x$ .

$- y^{2} - x^{2} + 14 x - 10 y - 64 = - 6$

Step 2.16

Reorder $- y^{2}$ and $- x^{2}$ .

$- x^{2} - y^{2} + 14 x - 10 y - 64 = - 6$

Step 3

Move all terms not containing a variable to the right side of the equation.

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Step 3.1

Add $64$ to both sides of the equation.

$- x^{2} - y^{2} + 14 x - 10 y = - 6 + 64$

Step 3.2

Add $- 6$ and $64$ .

$- x^{2} - y^{2} + 14 x - 10 y = 58$

Step 4

Divide both sides of the equation by $- 1$ .

$x^{2} + y^{2} - 14 x + 10 y = - 58$

Step 5

Complete the square for $x^{2} - 14 x$ .

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Step 5.1

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

$a = 1$

$b = - 14$

$c = 0$

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

$d = \frac{- 14}{2 \cdot 1}$

Step 5.3.2

Cancel the common factor of $- 14$ and $2$ .

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Step 5.3.2.1

Factor $2$ out of $- 14$ .

$d = \frac{2 \cdot - 7}{2 \cdot 1}$

Step 5.3.2.2

Cancel the common factors.

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Step 5.3.2.2.1

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

$d = \frac{2 \cdot - 7}{2 (1)}$

Step 5.3.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 7}{2 \cdot 1}$

Step 5.3.2.2.3

Rewrite the expression.

$d = \frac{- 7}{1}$

Step 5.3.2.2.4

Divide $- 7$ by $1$ .

$d = - 7$

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

$e = 0 - \frac{{(- 14)}^{2}}{4 \cdot 1}$

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 $- 14$ to the power of $2$ .

$e = 0 - \frac{196}{4 \cdot 1}$

Step 5.4.2.1.2

Multiply $4$ by $1$ .

$e = 0 - \frac{196}{4}$

Step 5.4.2.1.3

Divide $196$ by $4$ .

$e = 0 - 1 \cdot 49$

Step 5.4.2.1.4

Multiply $- 1$ by $49$ .

$e = 0 - 49$

Step 5.4.2.2

Subtract $49$ from $0$ .

$e = - 49$

Step 5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x - 7)}^{2} - 49$ .

${(x - 7)}^{2} - 49$

Step 6

Substitute ${(x - 7)}^{2} - 49$ for $x^{2} - 14 x$ in the equation $x^{2} + y^{2} - 14 x + 10 y = - 58$ .

${(x - 7)}^{2} - 49 + y^{2} + 10 y = - 58$

Step 7

Move $- 49$ to the right side of the equation by adding $49$ to both sides.

${(x - 7)}^{2} + y^{2} + 10 y = - 58 + 49$

Step 8

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

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Step 8.1

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 8.2

Consider the vertex form of a parabola.

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

Step 8.3

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

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Step 8.3.1

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

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

Step 8.3.2

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

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Step 8.3.2.1

Factor $2$ out of $10$ .

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

Step 8.3.2.2

Cancel the common factors.

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Step 8.3.2.2.1

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

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

Step 8.3.2.2.2

Cancel the common factor.

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

Step 8.3.2.2.3

Rewrite the expression.

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

Step 8.3.2.2.4

Divide $5$ by $1$ .

$d = 5$

Step 8.4

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

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Step 8.4.1

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 8.4.2

Simplify the right side.

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Step 8.4.2.1

Simplify each term.

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Step 8.4.2.1.1

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

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

Step 8.4.2.1.2

Multiply $4$ by $1$ .

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

Step 8.4.2.1.3

Divide $100$ by $4$ .

$e = 0 - 1 \cdot 25$

Step 8.4.2.1.4

Multiply $- 1$ by $25$ .

$e = 0 - 25$

Step 8.4.2.2

Subtract $25$ from $0$ .

$e = - 25$

Step 8.5

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

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

Step 9

Substitute ${(y + 5)}^{2} - 25$ for $y^{2} + 10 y$ in the equation $x^{2} + y^{2} - 14 x + 10 y = - 58$ .

${(x - 7)}^{2} + {(y + 5)}^{2} - 25 = - 58 + 49$

Step 10

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

${(x - 7)}^{2} + {(y + 5)}^{2} = - 58 + 49 + 25$

Step 11

Simplify $- 58 + 49 + 25$ .

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Step 11.1

Add $- 58$ and $49$ .

${(x - 7)}^{2} + {(y + 5)}^{2} = - 9 + 25$

Step 11.2

Add $- 9$ and $25$ .

${(x - 7)}^{2} + {(y + 5)}^{2} = 16$

Step 12

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 13

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.

$r = 4$

$h = 7$

$k = - 5$

Step 14

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

Center: $(7, - 5)$

Step 15

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

Center: $(7, - 5)$

Radius: $4$

Step 16