Trigonometry Examples

${(x + 3)}^{2} + {(y - 3)}^{2} = 53$

Step 1

Subtract ${(y - 3)}^{2}$ from both sides of the equation.

${(x + 3)}^{2} = 53 - {(y - 3)}^{2}$

Step 2

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

$x + 3 = \pm \sqrt{53 - {(y - 3)}^{2}}$

Step 3

Simplify $\pm \sqrt{53 - {(y - 3)}^{2}}$ .

Tap for more steps...

Step 3.1

Rewrite ${(y - 3)}^{2}$ as $(y - 3) (y - 3)$ .

$x + 3 = \pm \sqrt{53 - ((y - 3) (y - 3))}$

Step 3.2

Expand $(y - 3) (y - 3)$ using the FOIL Method.

Tap for more steps...

Step 3.2.1

Apply the distributive property.

$x + 3 = \pm \sqrt{53 - (y (y - 3) - 3 (y - 3))}$

Step 3.2.2

Apply the distributive property.

$x + 3 = \pm \sqrt{53 - (y \cdot y + y \cdot - 3 - 3 (y - 3))}$

Step 3.2.3

Apply the distributive property.

$x + 3 = \pm \sqrt{53 - (y \cdot y + y \cdot - 3 - 3 y - 3 \cdot - 3)}$

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 $y$ .

$x + 3 = \pm \sqrt{53 - (y^{2} + y \cdot - 3 - 3 y - 3 \cdot - 3)}$

Step 3.3.1.2

Move $- 3$ to the left of $y$ .

$x + 3 = \pm \sqrt{53 - (y^{2} - 3 \cdot y - 3 y - 3 \cdot - 3)}$

Step 3.3.1.3

Multiply $- 3$ by $- 3$ .

$x + 3 = \pm \sqrt{53 - (y^{2} - 3 y - 3 y + 9)}$

Step 3.3.2

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

$x + 3 = \pm \sqrt{53 - (y^{2} - 6 y + 9)}$

Step 3.4

Apply the distributive property.

$x + 3 = \pm \sqrt{53 - y^{2} - (- 6 y) - 1 \cdot 9}$

Step 3.5

Simplify.

Tap for more steps...

Step 3.5.1

Multiply $- 6$ by $- 1$ .

$x + 3 = \pm \sqrt{53 - y^{2} + 6 y - 1 \cdot 9}$

Step 3.5.2

Multiply $- 1$ by $9$ .

$x + 3 = \pm \sqrt{53 - y^{2} + 6 y - 9}$

Step 3.6

Subtract $9$ from $53$ .

$x + 3 = \pm \sqrt{- y^{2} + 6 y + 44}$

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.

$x + 3 = \sqrt{- y^{2} + 6 y + 44}$

Step 4.2

Subtract $3$ from both sides of the equation.

$x = \sqrt{- y^{2} + 6 y + 44} - 3$

Step 4.3

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

$x + 3 = - \sqrt{- y^{2} + 6 y + 44}$

Step 4.4

Subtract $3$ from both sides of the equation.

$x = - \sqrt{- y^{2} + 6 y + 44} - 3$

Step 4.5

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

$x = \sqrt{- y^{2} + 6 y + 44} - 3$

$x = - \sqrt{- y^{2} + 6 y + 44} - 3$

$x = \sqrt{- y^{2} + 6 y + 44} - 3$

$x = - \sqrt{- y^{2} + 6 y + 44} - 3$