Calculus Examples

${(x + 9)}^{2} + {(y - 12)}^{2} = 125$

Step 1

Subtract ${(y - 12)}^{2}$ from both sides of the equation.

${(x + 9)}^{2} = 125 - {(y - 12)}^{2}$

Step 2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x + 9 = \pm \sqrt{125 - {(y - 12)}^{2}}$

Step 3

Simplify $\pm \sqrt{125 - {(y - 12)}^{2}}$ .

Tap for more steps...

Step 3.1

Rewrite ${(y - 12)}^{2}$ as $(y - 12) (y - 12)$ .

$x + 9 = \pm \sqrt{125 - ((y - 12) (y - 12))}$

Step 3.2

Expand $(y - 12) (y - 12)$ using the FOIL Method.

Tap for more steps...

Step 3.2.1

Apply the distributive property.

$x + 9 = \pm \sqrt{125 - (y (y - 12) - 12 (y - 12))}$

Step 3.2.2

Apply the distributive property.

$x + 9 = \pm \sqrt{125 - (y \cdot y + y \cdot - 12 - 12 (y - 12))}$

Step 3.2.3

Apply the distributive property.

$x + 9 = \pm \sqrt{125 - (y \cdot y + y \cdot - 12 - 12 y - 12 \cdot - 12)}$

Step 3.3

Simplify and combine like terms.

Tap for more steps...

Step 3.3.1

Simplify each term.

Tap for more steps...

Step 3.3.1.1

Multiply $y$ by $y$ .

$x + 9 = \pm \sqrt{125 - (y^{2} + y \cdot - 12 - 12 y - 12 \cdot - 12)}$

Step 3.3.1.2

Move $- 12$ to the left of $y$ .

$x + 9 = \pm \sqrt{125 - (y^{2} - 12 \cdot y - 12 y - 12 \cdot - 12)}$

Step 3.3.1.3

Multiply $- 12$ by $- 12$ .

$x + 9 = \pm \sqrt{125 - (y^{2} - 12 y - 12 y + 144)}$

Step 3.3.2

Subtract $12 y$ from $- 12 y$ .

$x + 9 = \pm \sqrt{125 - (y^{2} - 24 y + 144)}$

Step 3.4

Apply the distributive property.

$x + 9 = \pm \sqrt{125 - y^{2} - (- 24 y) - 1 \cdot 144}$

Step 3.5

Simplify.

Tap for more steps...

Step 3.5.1

Multiply $- 24$ by $- 1$ .

$x + 9 = \pm \sqrt{125 - y^{2} + 24 y - 1 \cdot 144}$

Step 3.5.2

Multiply $- 1$ by $144$ .

$x + 9 = \pm \sqrt{125 - y^{2} + 24 y - 144}$

Step 3.6

Subtract $144$ from $125$ .

$x + 9 = \pm \sqrt{- y^{2} + 24 y - 19}$

Step 4

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 4.1

First, use the positive value of the $\pm$ to find the first solution.

$x + 9 = \sqrt{- y^{2} + 24 y - 19}$

Step 4.2

Subtract $9$ from both sides of the equation.

$x = \sqrt{- y^{2} + 24 y - 19} - 9$

Step 4.3

Next, use the negative value of the $\pm$ to find the second solution.

$x + 9 = - \sqrt{- y^{2} + 24 y - 19}$

Step 4.4

Subtract $9$ from both sides of the equation.

$x = - \sqrt{- y^{2} + 24 y - 19} - 9$

Step 4.5

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{- y^{2} + 24 y - 19} - 9$

$x = - \sqrt{- y^{2} + 24 y - 19} - 9$

$x = \sqrt{- y^{2} + 24 y - 19} - 9$

$x = - \sqrt{- y^{2} + 24 y - 19} - 9$