Solve for y (x+4)^2+y^2=22

${(x + 4)}^{2} + y^{2} = 22$

Subtract ${(x + 4)}^{2}$ from both sides of the equation.

$y^{2} = 22 - {(x + 4)}^{2}$

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

$y = \pm \sqrt{22 - {(x + 4)}^{2}}$

Simplify $\pm \sqrt{22 - {(x + 4)}^{2}}$ .

Rewrite ${(x + 4)}^{2}$ as $(x + 4) (x + 4)$ .

$y = \pm \sqrt{22 - ((x + 4) (x + 4))}$

Expand $(x + 4) (x + 4)$ using the FOIL Method.

Apply the distributive property.

$y = \pm \sqrt{22 - (x (x + 4) + 4 (x + 4))}$

Apply the distributive property.

$y = \pm \sqrt{22 - (x \cdot x + x \cdot 4 + 4 (x + 4))}$

Apply the distributive property.

$y = \pm \sqrt{22 - (x \cdot x + x \cdot 4 + 4 x + 4 \cdot 4)}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = \pm \sqrt{22 - (x^{2} + x \cdot 4 + 4 x + 4 \cdot 4)}$

Move $4$ to the left of $x$ .

$y = \pm \sqrt{22 - (x^{2} + 4 \cdot x + 4 x + 4 \cdot 4)}$

Multiply $4$ by $4$ .

$y = \pm \sqrt{22 - (x^{2} + 4 x + 4 x + 16)}$

Add $4 x$ and $4 x$ .

$y = \pm \sqrt{22 - (x^{2} + 8 x + 16)}$

Apply the distributive property.

$y = \pm \sqrt{22 - x^{2} - (8 x) - 1 \cdot 16}$

Simplify.

Multiply $8$ by $- 1$ .

$y = \pm \sqrt{22 - x^{2} - 8 x - 1 \cdot 16}$

Multiply $- 1$ by $16$ .

$y = \pm \sqrt{22 - x^{2} - 8 x - 16}$

Subtract $16$ from $22$ .

$y = \pm \sqrt{- x^{2} - 8 x + 6}$

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

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

$y = \sqrt{- x^{2} - 8 x + 6}$

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

$y = - \sqrt{- x^{2} - 8 x + 6}$

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

$y = \sqrt{- x^{2} - 8 x + 6}$

$y = - \sqrt{- x^{2} - 8 x + 6}$

$y = \sqrt{- x^{2} - 8 x + 6}$

$y = - \sqrt{- x^{2} - 8 x + 6}$