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

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

Step 1

Since

y

$y$ 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

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

Step 2

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

Tap for more steps...

Step 2.1

Subtract

x^{2}

$x^{2}$ 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

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

Step 2.2

Add

5 y^{2}

$5 y^{2}$ 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

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

Step 2.3

Add

4 x

$4 x$ 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

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

Step 2.4

Subtract

4 y

$4 y$ 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

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

Step 2.5

Subtract

30

$30$ from

9

$9$ .

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

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

Step 2.6

Add

- 6 y^{2}

$- 6 y^{2}$ and

5 y^{2}

$5 y^{2}$ .

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

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

Step 2.7

Add

- 4 y

$- 4 y$ and

y

$y$ .

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

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

Step 2.8

Subtract

3 y

$3 y$ from

- 3 y

$- 3 y$ .

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

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

Step 2.9

Subtract

4 y

$4 y$ from

- 6 y

$- 6 y$ .

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

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

Step 2.10

Add

- 21

$- 21$ and

17

$17$ .

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

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

Step 2.11

Add

10 x

$10 x$ and

4 x

$4 x$ .

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

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

Step 2.12

Subtract

60

$60$ from

- 4

$- 4$ .

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

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

Step 2.13

Move

- 64

$- 64$ .

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

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

Step 2.14

Move

- 10 y

$- 10 y$ .

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

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

Step 2.15

Move

14 x

$14 x$ .

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

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

Step 2.16

Reorder

- y^{2}

$- y^{2}$ and

- x^{2}

$- x^{2}$ .

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

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

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

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

Step 3

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

Tap for more steps...

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

$x^{2} - 14 x$ .

Tap for more steps...

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

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

Tap for more steps...

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

$2$ .

Tap for more steps...

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.

Tap for more steps...

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

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

Tap for more steps...

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.

Tap for more steps...

Step 5.4.2.1

Simplify each term.

Tap for more steps...

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

$y^{2} + 10 y$ .

Tap for more steps...

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

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

Tap for more steps...

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

$2$ .

Tap for more steps...

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.

Tap for more steps...

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

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

Tap for more steps...

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.

Tap for more steps...

Step 8.4.2.1

Simplify each term.

Tap for more steps...

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

$- 58 + 49 + 25$ .

Tap for more steps...

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