Algebra Examples

$x^{2} + y^{2} = 16$ $x^{2} - 4 x + y^{2} = 12$

Step 1

Solve for $x$ in $x^{2} + y^{2} = 16$ .

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Step 1.1

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

$x^{2} = 16 - y^{2}$

$x^{2} - 4 x + y^{2} = 12$

Step 1.2

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

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

$x^{2} - 4 x + y^{2} = 12$

Step 1.3

Simplify $\pm \sqrt{16 - y^{2}}$ .

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Step 1.3.1

Rewrite $16$ as $4^{2}$ .

$x = \pm \sqrt{4^{2} - y^{2}}$

$x^{2} - 4 x + y^{2} = 12$

Step 1.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4$ and $b = y$ .

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

$x^{2} - 4 x + y^{2} = 12$

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

$x^{2} - 4 x + y^{2} = 12$

Step 1.4

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

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Step 1.4.1

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

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

$x^{2} - 4 x + y^{2} = 12$

Step 1.4.2

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

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

$x^{2} - 4 x + y^{2} = 12$

Step 1.4.3

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

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

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

$x^{2} - 4 x + y^{2} = 12$

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

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

$x^{2} - 4 x + y^{2} = 12$

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

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

$x^{2} - 4 x + y^{2} = 12$

Step 2

Solve the system $x = \sqrt{(4 + y) (4 - y)}, x^{2} - 4 x + y^{2} = 12$ .

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Step 2.1

Replace all occurrences of $x$ with $\sqrt{(4 + y) (4 - y)}$ in each equation.

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Step 2.1.1

Replace all occurrences of $x$ in $x^{2} - 4 x + y^{2} = 12$ with $\sqrt{(4 + y) (4 - y)}$ .

${(\sqrt{(4 + y) (4 - y)})}^{2} - 4 \sqrt{(4 + y) (4 - y)} + y^{2} = 12$

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

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Simplify ${(\sqrt{(4 + y) (4 - y)})}^{2} - 4 \sqrt{(4 + y) (4 - y)} + y^{2}$ .

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Step 2.1.2.1.1

Simplify each term.

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Step 2.1.2.1.1.1

Rewrite ${\sqrt{(4 + y) (4 - y)}}^{2}$ as $(4 + y) (4 - y)$ .

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Step 2.1.2.1.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(4 + y) (4 - y)}$ as ${((4 + y) (4 - y))}^{\frac{1}{2}}$ .

${({((4 + y) (4 - y))}^{\frac{1}{2}})}^{2} - 4 \sqrt{(4 + y) (4 - y)} + y^{2} = 12$

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

Step 2.1.2.1.1.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${((4 + y) (4 - y))}^{\frac{1}{2} \cdot 2} - 4 \sqrt{(4 + y) (4 - y)} + y^{2} = 12$

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

Step 2.1.2.1.1.1.3

Combine $\frac{1}{2}$ and $2$ .

${((4 + y) (4 - y))}^{\frac{2}{2}} - 4 \sqrt{(4 + y) (4 - y)} + y^{2} = 12$

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

Step 2.1.2.1.1.1.4

Cancel the common factor of $2$ .

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Step 2.1.2.1.1.1.4.1

Cancel the common factor.

${((4 + y) (4 - y))}^{\frac{2}{2}} - 4 \sqrt{(4 + y) (4 - y)} + y^{2} = 12$

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

Step 2.1.2.1.1.1.4.2

Rewrite the expression.

$((4 + y) (4 - y)) - 4 \sqrt{(4 + y) (4 - y)} + y^{2} = 12$

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

$((4 + y) (4 - y)) - 4 \sqrt{(4 + y) (4 - y)} + y^{2} = 12$

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

Step 2.1.2.1.1.1.5

Simplify.

$(4 + y) (4 - y) - 4 \sqrt{(4 + y) (4 - y)} + y^{2} = 12$

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

$(4 + y) (4 - y) - 4 \sqrt{(4 + y) (4 - y)} + y^{2} = 12$

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

Step 2.1.2.1.1.2

Expand $(4 + y) (4 - y)$ using the FOIL Method.

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Step 2.1.2.1.1.2.1

Apply the distributive property.

$4 (4 - y) + y (4 - y) - 4 \sqrt{(4 + y) (4 - y)} + y^{2} = 12$

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

Step 2.1.2.1.1.2.2

Apply the distributive property.

$4 \cdot 4 + 4 (- y) + y (4 - y) - 4 \sqrt{(4 + y) (4 - y)} + y^{2} = 12$

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

Step 2.1.2.1.1.2.3

Apply the distributive property.

$4 \cdot 4 + 4 (- y) + y \cdot 4 + y (- y) - 4 \sqrt{(4 + y) (4 - y)} + y^{2} = 12$

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

$4 \cdot 4 + 4 (- y) + y \cdot 4 + y (- y) - 4 \sqrt{(4 + y) (4 - y)} + y^{2} = 12$

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

Step 2.1.2.1.1.3

Simplify and combine like terms.

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Step 2.1.2.1.1.3.1

Simplify each term.

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Step 2.1.2.1.1.3.1.1

Multiply $4$ by $4$ .

$16 + 4 (- y) + y \cdot 4 + y (- y) - 4 \sqrt{(4 + y) (4 - y)} + y^{2} = 12$

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

Step 2.1.2.1.1.3.1.2

Multiply $- 1$ by $4$ .

$16 - 4 y + y \cdot 4 + y (- y) - 4 \sqrt{(4 + y) (4 - y)} + y^{2} = 12$

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

Step 2.1.2.1.1.3.1.3

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

$16 - 4 y + 4 \cdot y + y (- y) - 4 \sqrt{(4 + y) (4 - y)} + y^{2} = 12$

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

Step 2.1.2.1.1.3.1.4

Rewrite using the commutative property of multiplication.

$16 - 4 y + 4 y - y \cdot y - 4 \sqrt{(4 + y) (4 - y)} + y^{2} = 12$

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

Step 2.1.2.1.1.3.1.5

Multiply $y$ by $y$ by adding the exponents.

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Step 2.1.2.1.1.3.1.5.1

Move $y$ .

$16 - 4 y + 4 y - (y \cdot y) - 4 \sqrt{(4 + y) (4 - y)} + y^{2} = 12$

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

Step 2.1.2.1.1.3.1.5.2

Multiply $y$ by $y$ .

$16 - 4 y + 4 y - y^{2} - 4 \sqrt{(4 + y) (4 - y)} + y^{2} = 12$

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

$16 - 4 y + 4 y - y^{2} - 4 \sqrt{(4 + y) (4 - y)} + y^{2} = 12$

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

$16 - 4 y + 4 y - y^{2} - 4 \sqrt{(4 + y) (4 - y)} + y^{2} = 12$

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

Step 2.1.2.1.1.3.2

Add $- 4 y$ and $4 y$ .

$16 + 0 - y^{2} - 4 \sqrt{(4 + y) (4 - y)} + y^{2} = 12$

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

Step 2.1.2.1.1.3.3

Add $16$ and $0$ .

$16 - y^{2} - 4 \sqrt{(4 + y) (4 - y)} + y^{2} = 12$

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

$16 - y^{2} - 4 \sqrt{(4 + y) (4 - y)} + y^{2} = 12$

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

$16 - y^{2} - 4 \sqrt{(4 + y) (4 - y)} + y^{2} = 12$

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

Step 2.1.2.1.2

Combine the opposite terms in $16 - y^{2} - 4 \sqrt{(4 + y) (4 - y)} + y^{2}$ .

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Step 2.1.2.1.2.1

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

$16 + 0 - 4 \sqrt{(4 + y) (4 - y)} = 12$

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

Step 2.1.2.1.2.2

Add $16$ and $0$ .

$16 - 4 \sqrt{(4 + y) (4 - y)} = 12$

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

$16 - 4 \sqrt{(4 + y) (4 - y)} = 12$

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

$16 - 4 \sqrt{(4 + y) (4 - y)} = 12$

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

$16 - 4 \sqrt{(4 + y) (4 - y)} = 12$

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

$16 - 4 \sqrt{(4 + y) (4 - y)} = 12$

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

Step 2.2

Solve for $y$ in $16 - 4 \sqrt{(4 + y) (4 - y)} = 12$ .

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Step 2.2.1

Move all terms not containing $- 4 \sqrt{(4 + y) (4 - y)}$ to the right side of the equation.

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Step 2.2.1.1

Subtract $16$ from both sides of the equation.

$- 4 \sqrt{(4 + y) (4 - y)} = 12 - 16$

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

Step 2.2.1.2

Subtract $16$ from $12$ .

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

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

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

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

Step 2.2.2

To remove the radical on the left side of the equation, square both sides of the equation.

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

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

Step 2.2.3

Simplify each side of the equation.

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Step 2.2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(4 + y) (4 - y)}$ as ${((4 + y) (4 - y))}^{\frac{1}{2}}$ .

${(- 4 {((4 + y) (4 - y))}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

Step 2.2.3.2

Simplify the left side.

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Step 2.2.3.2.1

Simplify ${(- 4 {((4 + y) (4 - y))}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.3.2.1.1

Expand $(4 + y) (4 - y)$ using the FOIL Method.

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Step 2.2.3.2.1.1.1

Apply the distributive property.

${(- 4 {(4 (4 - y) + y (4 - y))}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

Step 2.2.3.2.1.1.2

Apply the distributive property.

${(- 4 {(4 \cdot 4 + 4 (- y) + y (4 - y))}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

Step 2.2.3.2.1.1.3

Apply the distributive property.

${(- 4 {(4 \cdot 4 + 4 (- y) + y \cdot 4 + y (- y))}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

${(- 4 {(4 \cdot 4 + 4 (- y) + y \cdot 4 + y (- y))}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

Step 2.2.3.2.1.2

Simplify and combine like terms.

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Step 2.2.3.2.1.2.1

Simplify each term.

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Step 2.2.3.2.1.2.1.1

Multiply $4$ by $4$ .

${(- 4 {(16 + 4 (- y) + y \cdot 4 + y (- y))}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

Step 2.2.3.2.1.2.1.2

Multiply $- 1$ by $4$ .

${(- 4 {(16 - 4 y + y \cdot 4 + y (- y))}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

Step 2.2.3.2.1.2.1.3

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

${(- 4 {(16 - 4 y + 4 \cdot y + y (- y))}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

Step 2.2.3.2.1.2.1.4

Rewrite using the commutative property of multiplication.

${(- 4 {(16 - 4 y + 4 y - y \cdot y)}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

Step 2.2.3.2.1.2.1.5

Multiply $y$ by $y$ by adding the exponents.

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Step 2.2.3.2.1.2.1.5.1

Move $y$ .

${(- 4 {(16 - 4 y + 4 y - (y \cdot y))}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

Step 2.2.3.2.1.2.1.5.2

Multiply $y$ by $y$ .

${(- 4 {(16 - 4 y + 4 y - y^{2})}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

${(- 4 {(16 - 4 y + 4 y - y^{2})}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

${(- 4 {(16 - 4 y + 4 y - y^{2})}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

Step 2.2.3.2.1.2.2

Add $- 4 y$ and $4 y$ .

${(- 4 {(16 + 0 - y^{2})}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

Step 2.2.3.2.1.2.3

Add $16$ and $0$ .

${(- 4 {(16 - y^{2})}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

${(- 4 {(16 - y^{2})}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

Step 2.2.3.2.1.3

Apply the product rule to $- 4 {(16 - y^{2})}^{\frac{1}{2}}$ .

${(- 4)}^{2} {({(16 - y^{2})}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

Step 2.2.3.2.1.4

Raise $- 4$ to the power of $2$ .

$16 {({(16 - y^{2})}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

Step 2.2.3.2.1.5

Multiply the exponents in ${({(16 - y^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.3.2.1.5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$16 {(16 - y^{2})}^{\frac{1}{2} \cdot 2} = {(- 4)}^{2}$

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

Step 2.2.3.2.1.5.2

Cancel the common factor of $2$ .

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Step 2.2.3.2.1.5.2.1

Cancel the common factor.

$16 {(16 - y^{2})}^{\frac{1}{2} \cdot 2} = {(- 4)}^{2}$

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

Step 2.2.3.2.1.5.2.2

Rewrite the expression.

$16 (16 - y^{2}) = {(- 4)}^{2}$

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

$16 (16 - y^{2}) = {(- 4)}^{2}$

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

$16 (16 - y^{2}) = {(- 4)}^{2}$

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

Step 2.2.3.2.1.6

Simplify.

$16 (16 - y^{2}) = {(- 4)}^{2}$

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

Step 2.2.3.2.1.7

Apply the distributive property.

$16 \cdot 16 + 16 (- y^{2}) = {(- 4)}^{2}$

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

Step 2.2.3.2.1.8

Multiply.

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Step 2.2.3.2.1.8.1

Multiply $16$ by $16$ .

$256 + 16 (- y^{2}) = {(- 4)}^{2}$

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

Step 2.2.3.2.1.8.2

Multiply $- 1$ by $16$ .

$256 - 16 y^{2} = {(- 4)}^{2}$

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

$256 - 16 y^{2} = {(- 4)}^{2}$

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

$256 - 16 y^{2} = {(- 4)}^{2}$

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

$256 - 16 y^{2} = {(- 4)}^{2}$

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

Step 2.2.3.3

Simplify the right side.

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Step 2.2.3.3.1

Raise $- 4$ to the power of $2$ .

$256 - 16 y^{2} = 16$

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

$256 - 16 y^{2} = 16$

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

$256 - 16 y^{2} = 16$

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

Step 2.2.4

Solve for $y$ .

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Step 2.2.4.1

Move all terms not containing $y$ to the right side of the equation.

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Step 2.2.4.1.1

Subtract $256$ from both sides of the equation.

$- 16 y^{2} = 16 - 256$

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

Step 2.2.4.1.2

Subtract $256$ from $16$ .

$- 16 y^{2} = - 240$

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

$- 16 y^{2} = - 240$

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

Step 2.2.4.2

Divide each term in $- 16 y^{2} = - 240$ by $- 16$ and simplify.

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Step 2.2.4.2.1

Divide each term in $- 16 y^{2} = - 240$ by $- 16$ .

$\frac{- 16 y^{2}}{- 16} = \frac{- 240}{- 16}$

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

Step 2.2.4.2.2

Simplify the left side.

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Step 2.2.4.2.2.1

Cancel the common factor of $- 16$ .

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Step 2.2.4.2.2.1.1

Cancel the common factor.

$\frac{- 16 y^{2}}{- 16} = \frac{- 240}{- 16}$

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

Step 2.2.4.2.2.1.2

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{- 240}{- 16}$

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

$y^{2} = \frac{- 240}{- 16}$

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

$y^{2} = \frac{- 240}{- 16}$

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

Step 2.2.4.2.3

Simplify the right side.

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Step 2.2.4.2.3.1

Divide $- 240$ by $- 16$ .

$y^{2} = 15$

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

$y^{2} = 15$

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

$y^{2} = 15$

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

Step 2.2.4.3

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

$y = \pm \sqrt{15}$

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

Step 2.2.4.4

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

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Step 2.2.4.4.1

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

$y = \sqrt{15}$

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

Step 2.2.4.4.2

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

$y = - \sqrt{15}$

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

Step 2.2.4.4.3

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

$y = \sqrt{15}, - \sqrt{15}$

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

$y = \sqrt{15}, - \sqrt{15}$

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

$y = \sqrt{15}, - \sqrt{15}$

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

$y = \sqrt{15}, - \sqrt{15}$

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

Step 2.3

Replace all occurrences of $y$ with $\sqrt{15}$ in each equation.

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Step 2.3.1

Replace all occurrences of $y$ in $x = \sqrt{(4 + y) (4 - y)}$ with $\sqrt{15}$ .

$x = \sqrt{(4 + \sqrt{15}) (4 - (\sqrt{15}))}$

$y = \sqrt{15}$

Step 2.3.2

Simplify $x = \sqrt{(4 + \sqrt{15}) (4 - (\sqrt{15}))}$ .

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Step 2.3.2.1

Simplify the left side.

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Step 2.3.2.1.1

Remove parentheses.

$x = \sqrt{(4 + \sqrt{15}) (4 - (\sqrt{15}))}$

$y = \sqrt{15}$

$x = \sqrt{(4 + \sqrt{15}) (4 - (\sqrt{15}))}$

$y = \sqrt{15}$

Step 2.3.2.2

Simplify the right side.

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Step 2.3.2.2.1

Simplify $\sqrt{(4 + \sqrt{15}) (4 - (\sqrt{15}))}$ .

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Step 2.3.2.2.1.1

Expand $(4 + \sqrt{15}) (4 - \sqrt{15})$ using the FOIL Method.

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Step 2.3.2.2.1.1.1

Apply the distributive property.

$x = \sqrt{4 (4 - \sqrt{15}) + \sqrt{15} (4 - \sqrt{15})}$

$y = \sqrt{15}$

Step 2.3.2.2.1.1.2

Apply the distributive property.

$x = \sqrt{4 \cdot 4 + 4 (- \sqrt{15}) + \sqrt{15} (4 - \sqrt{15})}$

$y = \sqrt{15}$

Step 2.3.2.2.1.1.3

Apply the distributive property.

$x = \sqrt{4 \cdot 4 + 4 (- \sqrt{15}) + \sqrt{15} \cdot 4 + \sqrt{15} (- \sqrt{15})}$

$y = \sqrt{15}$

$x = \sqrt{4 \cdot 4 + 4 (- \sqrt{15}) + \sqrt{15} \cdot 4 + \sqrt{15} (- \sqrt{15})}$

$y = \sqrt{15}$

Step 2.3.2.2.1.2

Simplify and combine like terms.

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Step 2.3.2.2.1.2.1

Simplify each term.

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Step 2.3.2.2.1.2.1.1

Multiply $4$ by $4$ .

$x = \sqrt{16 + 4 (- \sqrt{15}) + \sqrt{15} \cdot 4 + \sqrt{15} (- \sqrt{15})}$

$y = \sqrt{15}$

Step 2.3.2.2.1.2.1.2

Multiply $- 1$ by $4$ .

$x = \sqrt{16 - 4 \sqrt{15} + \sqrt{15} \cdot 4 + \sqrt{15} (- \sqrt{15})}$

$y = \sqrt{15}$

Step 2.3.2.2.1.2.1.3

Move $4$ to the left of $\sqrt{15}$ .

$x = \sqrt{16 - 4 \sqrt{15} + 4 \cdot \sqrt{15} + \sqrt{15} (- \sqrt{15})}$

$y = \sqrt{15}$

Step 2.3.2.2.1.2.1.4

Multiply $\sqrt{15} (- \sqrt{15})$ .

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Step 2.3.2.2.1.2.1.4.1

Raise $\sqrt{15}$ to the power of $1$ .

$x = \sqrt{16 - 4 \sqrt{15} + 4 \sqrt{15} - (\sqrt{15} \sqrt{15})}$

$y = \sqrt{15}$

Step 2.3.2.2.1.2.1.4.2

Raise $\sqrt{15}$ to the power of $1$ .

$x = \sqrt{16 - 4 \sqrt{15} + 4 \sqrt{15} - (\sqrt{15} \sqrt{15})}$

$y = \sqrt{15}$

Step 2.3.2.2.1.2.1.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \sqrt{16 - 4 \sqrt{15} + 4 \sqrt{15} - {\sqrt{15}}^{1 + 1}}$

$y = \sqrt{15}$

Step 2.3.2.2.1.2.1.4.4

Add $1$ and $1$ .

$x = \sqrt{16 - 4 \sqrt{15} + 4 \sqrt{15} - {\sqrt{15}}^{2}}$

$y = \sqrt{15}$

$x = \sqrt{16 - 4 \sqrt{15} + 4 \sqrt{15} - {\sqrt{15}}^{2}}$

$y = \sqrt{15}$

Step 2.3.2.2.1.2.1.5

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 2.3.2.2.1.2.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$x = \sqrt{16 - 4 \sqrt{15} + 4 \sqrt{15} - {(15^{\frac{1}{2}})}^{2}}$

$y = \sqrt{15}$

Step 2.3.2.2.1.2.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \sqrt{16 - 4 \sqrt{15} + 4 \sqrt{15} - 15^{\frac{1}{2} \cdot 2}}$

$y = \sqrt{15}$

Step 2.3.2.2.1.2.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$x = \sqrt{16 - 4 \sqrt{15} + 4 \sqrt{15} - 15^{\frac{2}{2}}}$

$y = \sqrt{15}$

Step 2.3.2.2.1.2.1.5.4

Cancel the common factor of $2$ .

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Step 2.3.2.2.1.2.1.5.4.1

Cancel the common factor.

$x = \sqrt{16 - 4 \sqrt{15} + 4 \sqrt{15} - 15^{\frac{2}{2}}}$

$y = \sqrt{15}$

Step 2.3.2.2.1.2.1.5.4.2

Rewrite the expression.

$x = \sqrt{16 - 4 \sqrt{15} + 4 \sqrt{15} - 15}$

$y = \sqrt{15}$

$x = \sqrt{16 - 4 \sqrt{15} + 4 \sqrt{15} - 15}$

$y = \sqrt{15}$

Step 2.3.2.2.1.2.1.5.5

Evaluate the exponent.

$x = \sqrt{16 - 4 \sqrt{15} + 4 \sqrt{15} - 1 \cdot 15}$

$y = \sqrt{15}$

$x = \sqrt{16 - 4 \sqrt{15} + 4 \sqrt{15} - 1 \cdot 15}$

$y = \sqrt{15}$

Step 2.3.2.2.1.2.1.6

Multiply $- 1$ by $15$ .

$x = \sqrt{16 - 4 \sqrt{15} + 4 \sqrt{15} - 15}$

$y = \sqrt{15}$

$x = \sqrt{16 - 4 \sqrt{15} + 4 \sqrt{15} - 15}$

$y = \sqrt{15}$

Step 2.3.2.2.1.2.2

Subtract $15$ from $16$ .

$x = \sqrt{1 - 4 \sqrt{15} + 4 \sqrt{15}}$

$y = \sqrt{15}$

Step 2.3.2.2.1.2.3

Add $- 4 \sqrt{15}$ and $4 \sqrt{15}$ .

$x = \sqrt{1 + 0}$

$y = \sqrt{15}$

Step 2.3.2.2.1.2.4

Add $1$ and $0$ .

$x = \sqrt{1}$

$y = \sqrt{15}$

$x = \sqrt{1}$

$y = \sqrt{15}$

Step 2.3.2.2.1.3

Any root of $1$ is $1$ .

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Step 2.3.2.2.1.3.1

Simplify each term.

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Step 2.3.2.2.1.3.1.1

Multiply $4$ by $4$ .

$x = 16 + 4 (- \sqrt{15}) + \sqrt{15} \cdot 4 + \sqrt{15} (- \sqrt{15})$

$y = \sqrt{15}$

Step 2.3.2.2.1.3.1.2

Multiply $- 1$ by $4$ .

$x = 16 - 4 \sqrt{15} + \sqrt{15} \cdot 4 + \sqrt{15} (- \sqrt{15})$

$y = \sqrt{15}$

Step 2.3.2.2.1.3.1.3

Move $4$ to the left of $\sqrt{15}$ .

$x = 16 - 4 \sqrt{15} + 4 \cdot \sqrt{15} + \sqrt{15} (- \sqrt{15})$

$y = \sqrt{15}$

Step 2.3.2.2.1.3.1.4

Multiply $\sqrt{15} (- \sqrt{15})$ .

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Step 2.3.2.2.1.3.1.4.1

Raise $\sqrt{15}$ to the power of $1$ .

$x = 16 - 4 \sqrt{15} + 4 \sqrt{15} - (\sqrt{15} \sqrt{15})$

$y = \sqrt{15}$

Step 2.3.2.2.1.3.1.4.2

Raise $\sqrt{15}$ to the power of $1$ .

$x = 16 - 4 \sqrt{15} + 4 \sqrt{15} - (\sqrt{15} \sqrt{15})$

$y = \sqrt{15}$

Step 2.3.2.2.1.3.1.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = 16 - 4 \sqrt{15} + 4 \sqrt{15} - {\sqrt{15}}^{1 + 1}$

$y = \sqrt{15}$

Step 2.3.2.2.1.3.1.4.4

Add $1$ and $1$ .

$x = 16 - 4 \sqrt{15} + 4 \sqrt{15} - {\sqrt{15}}^{2}$

$y = \sqrt{15}$

$x = 16 - 4 \sqrt{15} + 4 \sqrt{15} - {\sqrt{15}}^{2}$

$y = \sqrt{15}$

Step 2.3.2.2.1.3.1.5

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 2.3.2.2.1.3.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$x = 16 - 4 \sqrt{15} + 4 \sqrt{15} - {(15^{\frac{1}{2}})}^{2}$

$y = \sqrt{15}$

Step 2.3.2.2.1.3.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = 16 - 4 \sqrt{15} + 4 \sqrt{15} - 15^{\frac{1}{2} \cdot 2}$

$y = \sqrt{15}$

Step 2.3.2.2.1.3.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$x = 16 - 4 \sqrt{15} + 4 \sqrt{15} - 15^{\frac{2}{2}}$

$y = \sqrt{15}$

Step 2.3.2.2.1.3.1.5.4

Cancel the common factor of $2$ .

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Step 2.3.2.2.1.3.1.5.4.1

Cancel the common factor.

$x = 16 - 4 \sqrt{15} + 4 \sqrt{15} - 15^{\frac{2}{2}}$

$y = \sqrt{15}$

Step 2.3.2.2.1.3.1.5.4.2

Rewrite the expression.

$x = 16 - 4 \sqrt{15} + 4 \sqrt{15} - 15$

$y = \sqrt{15}$

$x = 16 - 4 \sqrt{15} + 4 \sqrt{15} - 15$

$y = \sqrt{15}$

Step 2.3.2.2.1.3.1.5.5

Evaluate the exponent.

$x = 16 - 4 \sqrt{15} + 4 \sqrt{15} - 1 \cdot 15$

$y = \sqrt{15}$

$x = 16 - 4 \sqrt{15} + 4 \sqrt{15} - 1 \cdot 15$

$y = \sqrt{15}$

Step 2.3.2.2.1.3.1.6

Multiply $- 1$ by $15$ .

$x = 16 - 4 \sqrt{15} + 4 \sqrt{15} - 15$

$y = \sqrt{15}$

$x = 16 - 4 \sqrt{15} + 4 \sqrt{15} - 15$

$y = \sqrt{15}$

Step 2.3.2.2.1.3.2

Subtract $15$ from $16$ .

$x = 1 - 4 \sqrt{15} + 4 \sqrt{15}$

$y = \sqrt{15}$

Step 2.3.2.2.1.3.3

Add $- 4 \sqrt{15}$ and $4 \sqrt{15}$ .

$x = 1 + 0$

$y = \sqrt{15}$

Step 2.3.2.2.1.3.4

Add $1$ and $0$ .

$x = 1$

$y = \sqrt{15}$

$x = 1$

$y = \sqrt{15}$

$x = 1$

$y = \sqrt{15}$

$x = 1$

$y = \sqrt{15}$

$x = 1$

$y = \sqrt{15}$

$x = 1$

$y = \sqrt{15}$

Step 2.4

Replace all occurrences of $y$ with $- \sqrt{15}$ in each equation.

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Step 2.4.1

Replace all occurrences of $y$ in $x = \sqrt{(4 + y) (4 - y)}$ with $- \sqrt{15}$ .

$x = \sqrt{(4 - \sqrt{15}) (4 - (- \sqrt{15}))}$

$y = - \sqrt{15}$

Step 2.4.2

Simplify $x = \sqrt{(4 - \sqrt{15}) (4 - (- \sqrt{15}))}$ .

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Step 2.4.2.1

Simplify the left side.

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Step 2.4.2.1.1

Remove parentheses.

$x = \sqrt{(4 - \sqrt{15}) (4 - (- \sqrt{15}))}$

$y = - \sqrt{15}$

$x = \sqrt{(4 - \sqrt{15}) (4 - (- \sqrt{15}))}$

$y = - \sqrt{15}$

Step 2.4.2.2

Simplify the right side.

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Step 2.4.2.2.1

Simplify $\sqrt{(4 - \sqrt{15}) (4 - (- \sqrt{15}))}$ .

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Step 2.4.2.2.1.1

Multiply $- - \sqrt{15}$ .

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Step 2.4.2.2.1.1.1

Multiply $- 1$ by $- 1$ .

$x = \sqrt{(4 - \sqrt{15}) (4 + 1 \sqrt{15})}$

$y = - \sqrt{15}$

Step 2.4.2.2.1.1.2

Multiply $\sqrt{15}$ by $1$ .

$x = \sqrt{(4 - \sqrt{15}) (4 + \sqrt{15})}$

$y = - \sqrt{15}$

$x = \sqrt{(4 - \sqrt{15}) (4 + \sqrt{15})}$

$y = - \sqrt{15}$

Step 2.4.2.2.1.2

Expand $(4 - \sqrt{15}) (4 + \sqrt{15})$ using the FOIL Method.

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Step 2.4.2.2.1.2.1

Apply the distributive property.

$x = \sqrt{4 (4 + \sqrt{15}) - \sqrt{15} (4 + \sqrt{15})}$

$y = - \sqrt{15}$

Step 2.4.2.2.1.2.2

Apply the distributive property.

$x = \sqrt{4 \cdot 4 + 4 \sqrt{15} - \sqrt{15} (4 + \sqrt{15})}$

$y = - \sqrt{15}$

Step 2.4.2.2.1.2.3

Apply the distributive property.

$x = \sqrt{4 \cdot 4 + 4 \sqrt{15} - \sqrt{15} \cdot 4 - \sqrt{15} \sqrt{15}}$

$y = - \sqrt{15}$

$x = \sqrt{4 \cdot 4 + 4 \sqrt{15} - \sqrt{15} \cdot 4 - \sqrt{15} \sqrt{15}}$

$y = - \sqrt{15}$

Step 2.4.2.2.1.3

Simplify and combine like terms.

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Step 2.4.2.2.1.3.1

Simplify each term.

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Step 2.4.2.2.1.3.1.1

Multiply $4$ by $4$ .

$x = \sqrt{16 + 4 \sqrt{15} - \sqrt{15} \cdot 4 - \sqrt{15} \sqrt{15}}$

$y = - \sqrt{15}$

Step 2.4.2.2.1.3.1.2

Multiply $4$ by $- 1$ .

$x = \sqrt{16 + 4 \sqrt{15} - 4 \sqrt{15} - \sqrt{15} \sqrt{15}}$

$y = - \sqrt{15}$

Step 2.4.2.2.1.3.1.3

Multiply $- \sqrt{15} \sqrt{15}$ .

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Step 2.4.2.2.1.3.1.3.1

Raise $\sqrt{15}$ to the power of $1$ .

$x = \sqrt{16 + 4 \sqrt{15} - 4 \sqrt{15} - (\sqrt{15} \sqrt{15})}$

$y = - \sqrt{15}$

Step 2.4.2.2.1.3.1.3.2

Raise $\sqrt{15}$ to the power of $1$ .

$x = \sqrt{16 + 4 \sqrt{15} - 4 \sqrt{15} - (\sqrt{15} \sqrt{15})}$

$y = - \sqrt{15}$

Step 2.4.2.2.1.3.1.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \sqrt{16 + 4 \sqrt{15} - 4 \sqrt{15} - {\sqrt{15}}^{1 + 1}}$

$y = - \sqrt{15}$

Step 2.4.2.2.1.3.1.3.4

Add $1$ and $1$ .

$x = \sqrt{16 + 4 \sqrt{15} - 4 \sqrt{15} - {\sqrt{15}}^{2}}$

$y = - \sqrt{15}$

$x = \sqrt{16 + 4 \sqrt{15} - 4 \sqrt{15} - {\sqrt{15}}^{2}}$

$y = - \sqrt{15}$

Step 2.4.2.2.1.3.1.4

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 2.4.2.2.1.3.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$x = \sqrt{16 + 4 \sqrt{15} - 4 \sqrt{15} - {(15^{\frac{1}{2}})}^{2}}$

$y = - \sqrt{15}$

Step 2.4.2.2.1.3.1.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \sqrt{16 + 4 \sqrt{15} - 4 \sqrt{15} - 15^{\frac{1}{2} \cdot 2}}$

$y = - \sqrt{15}$

Step 2.4.2.2.1.3.1.4.3

Combine $\frac{1}{2}$ and $2$ .

$x = \sqrt{16 + 4 \sqrt{15} - 4 \sqrt{15} - 15^{\frac{2}{2}}}$

$y = - \sqrt{15}$

Step 2.4.2.2.1.3.1.4.4

Cancel the common factor of $2$ .

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Step 2.4.2.2.1.3.1.4.4.1

Cancel the common factor.

$x = \sqrt{16 + 4 \sqrt{15} - 4 \sqrt{15} - 15^{\frac{2}{2}}}$

$y = - \sqrt{15}$

Step 2.4.2.2.1.3.1.4.4.2

Rewrite the expression.

$x = \sqrt{16 + 4 \sqrt{15} - 4 \sqrt{15} - 15}$

$y = - \sqrt{15}$

$x = \sqrt{16 + 4 \sqrt{15} - 4 \sqrt{15} - 15}$

$y = - \sqrt{15}$

Step 2.4.2.2.1.3.1.4.5

Evaluate the exponent.

$x = \sqrt{16 + 4 \sqrt{15} - 4 \sqrt{15} - 1 \cdot 15}$

$y = - \sqrt{15}$

$x = \sqrt{16 + 4 \sqrt{15} - 4 \sqrt{15} - 1 \cdot 15}$

$y = - \sqrt{15}$

Step 2.4.2.2.1.3.1.5

Multiply $- 1$ by $15$ .

$x = \sqrt{16 + 4 \sqrt{15} - 4 \sqrt{15} - 15}$

$y = - \sqrt{15}$

$x = \sqrt{16 + 4 \sqrt{15} - 4 \sqrt{15} - 15}$

$y = - \sqrt{15}$

Step 2.4.2.2.1.3.2

Subtract $15$ from $16$ .

$x = \sqrt{1 + 4 \sqrt{15} - 4 \sqrt{15}}$

$y = - \sqrt{15}$

Step 2.4.2.2.1.3.3

Subtract $4 \sqrt{15}$ from $4 \sqrt{15}$ .

$x = \sqrt{1 + 0}$

$y = - \sqrt{15}$

Step 2.4.2.2.1.3.4

Add $1$ and $0$ .

$x = \sqrt{1}$

$y = - \sqrt{15}$

$x = \sqrt{1}$

$y = - \sqrt{15}$

Step 2.4.2.2.1.4

Any root of $1$ is $1$ .

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Step 2.4.2.2.1.4.1

Simplify each term.

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Step 2.4.2.2.1.4.1.1

Multiply $4$ by $4$ .

$x = 16 + 4 \sqrt{15} - \sqrt{15} \cdot 4 - \sqrt{15} \sqrt{15}$

$y = - \sqrt{15}$

Step 2.4.2.2.1.4.1.2

Multiply $4$ by $- 1$ .

$x = 16 + 4 \sqrt{15} - 4 \sqrt{15} - \sqrt{15} \sqrt{15}$

$y = - \sqrt{15}$

Step 2.4.2.2.1.4.1.3

Multiply $- \sqrt{15} \sqrt{15}$ .

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Step 2.4.2.2.1.4.1.3.1

Raise $\sqrt{15}$ to the power of $1$ .

$x = 16 + 4 \sqrt{15} - 4 \sqrt{15} - (\sqrt{15} \sqrt{15})$

$y = - \sqrt{15}$

Step 2.4.2.2.1.4.1.3.2

Raise $\sqrt{15}$ to the power of $1$ .

$x = 16 + 4 \sqrt{15} - 4 \sqrt{15} - (\sqrt{15} \sqrt{15})$

$y = - \sqrt{15}$

Step 2.4.2.2.1.4.1.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = 16 + 4 \sqrt{15} - 4 \sqrt{15} - {\sqrt{15}}^{1 + 1}$

$y = - \sqrt{15}$

Step 2.4.2.2.1.4.1.3.4

Add $1$ and $1$ .

$x = 16 + 4 \sqrt{15} - 4 \sqrt{15} - {\sqrt{15}}^{2}$

$y = - \sqrt{15}$

$x = 16 + 4 \sqrt{15} - 4 \sqrt{15} - {\sqrt{15}}^{2}$

$y = - \sqrt{15}$

Step 2.4.2.2.1.4.1.4

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 2.4.2.2.1.4.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$x = 16 + 4 \sqrt{15} - 4 \sqrt{15} - {(15^{\frac{1}{2}})}^{2}$

$y = - \sqrt{15}$

Step 2.4.2.2.1.4.1.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = 16 + 4 \sqrt{15} - 4 \sqrt{15} - 15^{\frac{1}{2} \cdot 2}$

$y = - \sqrt{15}$

Step 2.4.2.2.1.4.1.4.3

Combine $\frac{1}{2}$ and $2$ .

$x = 16 + 4 \sqrt{15} - 4 \sqrt{15} - 15^{\frac{2}{2}}$

$y = - \sqrt{15}$

Step 2.4.2.2.1.4.1.4.4

Cancel the common factor of $2$ .

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Step 2.4.2.2.1.4.1.4.4.1

Cancel the common factor.

$x = 16 + 4 \sqrt{15} - 4 \sqrt{15} - 15^{\frac{2}{2}}$

$y = - \sqrt{15}$

Step 2.4.2.2.1.4.1.4.4.2

Rewrite the expression.

$x = 16 + 4 \sqrt{15} - 4 \sqrt{15} - 15$

$y = - \sqrt{15}$

$x = 16 + 4 \sqrt{15} - 4 \sqrt{15} - 15$

$y = - \sqrt{15}$

Step 2.4.2.2.1.4.1.4.5

Evaluate the exponent.

$x = 16 + 4 \sqrt{15} - 4 \sqrt{15} - 1 \cdot 15$

$y = - \sqrt{15}$

$x = 16 + 4 \sqrt{15} - 4 \sqrt{15} - 1 \cdot 15$

$y = - \sqrt{15}$

Step 2.4.2.2.1.4.1.5

Multiply $- 1$ by $15$ .

$x = 16 + 4 \sqrt{15} - 4 \sqrt{15} - 15$

$y = - \sqrt{15}$

$x = 16 + 4 \sqrt{15} - 4 \sqrt{15} - 15$

$y = - \sqrt{15}$

Step 2.4.2.2.1.4.2

Subtract $15$ from $16$ .

$x = 1 + 4 \sqrt{15} - 4 \sqrt{15}$

$y = - \sqrt{15}$

Step 2.4.2.2.1.4.3

Subtract $4 \sqrt{15}$ from $4 \sqrt{15}$ .

$x = 1 + 0$

$y = - \sqrt{15}$

Step 2.4.2.2.1.4.4

Add $1$ and $0$ .

$x = 1$

$y = - \sqrt{15}$

$x = 1$

$y = - \sqrt{15}$

$x = 1$

$y = - \sqrt{15}$

$x = 1$

$y = - \sqrt{15}$

$x = 1$

$y = - \sqrt{15}$

$x = 1$

$y = - \sqrt{15}$

$x = 1$

$y = - \sqrt{15}$

Step 3

Solve the system $x = - \sqrt{(4 + y) (4 - y)}, x^{2} - 4 x + y^{2} = 12$ .

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Step 3.1

Replace all occurrences of $x$ with $- \sqrt{(4 + y) (4 - y)}$ in each equation.

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Step 3.1.1

Replace all occurrences of $x$ in $x^{2} - 4 x + y^{2} = 12$ with $- \sqrt{(4 + y) (4 - y)}$ .

${(- \sqrt{(4 + y) (4 - y)})}^{2} - 4 (- \sqrt{(4 + y) (4 - y)}) + y^{2} = 12$

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

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Simplify ${(- \sqrt{(4 + y) (4 - y)})}^{2} - 4 (- \sqrt{(4 + y) (4 - y)}) + y^{2}$ .

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Step 3.1.2.1.1

Simplify each term.

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Step 3.1.2.1.1.1

Apply the product rule to $- \sqrt{(4 + y) (4 - y)}$ .

${(- 1)}^{2} {\sqrt{(4 + y) (4 - y)}}^{2} - 4 (- \sqrt{(4 + y) (4 - y)}) + y^{2} = 12$

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

Step 3.1.2.1.1.2

Raise $- 1$ to the power of $2$ .

$1 {\sqrt{(4 + y) (4 - y)}}^{2} - 4 (- \sqrt{(4 + y) (4 - y)}) + y^{2} = 12$

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

Step 3.1.2.1.1.3

Multiply ${\sqrt{(4 + y) (4 - y)}}^{2}$ by $1$ .

${\sqrt{(4 + y) (4 - y)}}^{2} - 4 (- \sqrt{(4 + y) (4 - y)}) + y^{2} = 12$

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

Step 3.1.2.1.1.4

Rewrite ${\sqrt{(4 + y) (4 - y)}}^{2}$ as $(4 + y) (4 - y)$ .

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Step 3.1.2.1.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(4 + y) (4 - y)}$ as ${((4 + y) (4 - y))}^{\frac{1}{2}}$ .

${({((4 + y) (4 - y))}^{\frac{1}{2}})}^{2} - 4 (- \sqrt{(4 + y) (4 - y)}) + y^{2} = 12$

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

Step 3.1.2.1.1.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${((4 + y) (4 - y))}^{\frac{1}{2} \cdot 2} - 4 (- \sqrt{(4 + y) (4 - y)}) + y^{2} = 12$

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

Step 3.1.2.1.1.4.3

Combine $\frac{1}{2}$ and $2$ .

${((4 + y) (4 - y))}^{\frac{2}{2}} - 4 (- \sqrt{(4 + y) (4 - y)}) + y^{2} = 12$

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

Step 3.1.2.1.1.4.4

Cancel the common factor of $2$ .

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Step 3.1.2.1.1.4.4.1

Cancel the common factor.

${((4 + y) (4 - y))}^{\frac{2}{2}} - 4 (- \sqrt{(4 + y) (4 - y)}) + y^{2} = 12$

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

Step 3.1.2.1.1.4.4.2

Rewrite the expression.

$((4 + y) (4 - y)) - 4 (- \sqrt{(4 + y) (4 - y)}) + y^{2} = 12$

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

$((4 + y) (4 - y)) - 4 (- \sqrt{(4 + y) (4 - y)}) + y^{2} = 12$

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

Step 3.1.2.1.1.4.5

Simplify.

$(4 + y) (4 - y) - 4 (- \sqrt{(4 + y) (4 - y)}) + y^{2} = 12$

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

$(4 + y) (4 - y) - 4 (- \sqrt{(4 + y) (4 - y)}) + y^{2} = 12$

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

Step 3.1.2.1.1.5

Expand $(4 + y) (4 - y)$ using the FOIL Method.

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Step 3.1.2.1.1.5.1

Apply the distributive property.

$4 (4 - y) + y (4 - y) - 4 (- \sqrt{(4 + y) (4 - y)}) + y^{2} = 12$

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

Step 3.1.2.1.1.5.2

Apply the distributive property.

$4 \cdot 4 + 4 (- y) + y (4 - y) - 4 (- \sqrt{(4 + y) (4 - y)}) + y^{2} = 12$

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

Step 3.1.2.1.1.5.3

Apply the distributive property.

$4 \cdot 4 + 4 (- y) + y \cdot 4 + y (- y) - 4 (- \sqrt{(4 + y) (4 - y)}) + y^{2} = 12$

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

$4 \cdot 4 + 4 (- y) + y \cdot 4 + y (- y) - 4 (- \sqrt{(4 + y) (4 - y)}) + y^{2} = 12$

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

Step 3.1.2.1.1.6

Simplify and combine like terms.

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Step 3.1.2.1.1.6.1

Simplify each term.

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Step 3.1.2.1.1.6.1.1

Multiply $4$ by $4$ .

$16 + 4 (- y) + y \cdot 4 + y (- y) - 4 (- \sqrt{(4 + y) (4 - y)}) + y^{2} = 12$

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

Step 3.1.2.1.1.6.1.2

Multiply $- 1$ by $4$ .

$16 - 4 y + y \cdot 4 + y (- y) - 4 (- \sqrt{(4 + y) (4 - y)}) + y^{2} = 12$

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

Step 3.1.2.1.1.6.1.3

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

$16 - 4 y + 4 \cdot y + y (- y) - 4 (- \sqrt{(4 + y) (4 - y)}) + y^{2} = 12$

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

Step 3.1.2.1.1.6.1.4

Rewrite using the commutative property of multiplication.

$16 - 4 y + 4 y - y \cdot y - 4 (- \sqrt{(4 + y) (4 - y)}) + y^{2} = 12$

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

Step 3.1.2.1.1.6.1.5

Multiply $y$ by $y$ by adding the exponents.

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Step 3.1.2.1.1.6.1.5.1

Move $y$ .

$16 - 4 y + 4 y - (y \cdot y) - 4 (- \sqrt{(4 + y) (4 - y)}) + y^{2} = 12$

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

Step 3.1.2.1.1.6.1.5.2

Multiply $y$ by $y$ .

$16 - 4 y + 4 y - y^{2} - 4 (- \sqrt{(4 + y) (4 - y)}) + y^{2} = 12$

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

$16 - 4 y + 4 y - y^{2} - 4 (- \sqrt{(4 + y) (4 - y)}) + y^{2} = 12$

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

$16 - 4 y + 4 y - y^{2} - 4 (- \sqrt{(4 + y) (4 - y)}) + y^{2} = 12$

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

Step 3.1.2.1.1.6.2

Add $- 4 y$ and $4 y$ .

$16 + 0 - y^{2} - 4 (- \sqrt{(4 + y) (4 - y)}) + y^{2} = 12$

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

Step 3.1.2.1.1.6.3

Add $16$ and $0$ .

$16 - y^{2} - 4 (- \sqrt{(4 + y) (4 - y)}) + y^{2} = 12$

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

$16 - y^{2} - 4 (- \sqrt{(4 + y) (4 - y)}) + y^{2} = 12$

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

Step 3.1.2.1.1.7

Multiply $- 1$ by $- 4$ .

$16 - y^{2} + 4 \sqrt{(4 + y) (4 - y)} + y^{2} = 12$

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

$16 - y^{2} + 4 \sqrt{(4 + y) (4 - y)} + y^{2} = 12$

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

Step 3.1.2.1.2

Combine the opposite terms in $16 - y^{2} + 4 \sqrt{(4 + y) (4 - y)} + y^{2}$ .

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Step 3.1.2.1.2.1

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

$16 + 0 + 4 \sqrt{(4 + y) (4 - y)} = 12$

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

Step 3.1.2.1.2.2

Add $16$ and $0$ .

$16 + 4 \sqrt{(4 + y) (4 - y)} = 12$

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

$16 + 4 \sqrt{(4 + y) (4 - y)} = 12$

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

$16 + 4 \sqrt{(4 + y) (4 - y)} = 12$

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

$16 + 4 \sqrt{(4 + y) (4 - y)} = 12$

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

$16 + 4 \sqrt{(4 + y) (4 - y)} = 12$

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

Step 3.2

Solve for $y$ in $16 + 4 \sqrt{(4 + y) (4 - y)} = 12$ .

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Step 3.2.1

Move all terms not containing $4 \sqrt{(4 + y) (4 - y)}$ to the right side of the equation.

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Step 3.2.1.1

Subtract $16$ from both sides of the equation.

$4 \sqrt{(4 + y) (4 - y)} = 12 - 16$

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

Step 3.2.1.2

Subtract $16$ from $12$ .

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

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

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

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

Step 3.2.2

To remove the radical on the left side of the equation, square both sides of the equation.

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

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

Step 3.2.3

Simplify each side of the equation.

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Step 3.2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(4 + y) (4 - y)}$ as ${((4 + y) (4 - y))}^{\frac{1}{2}}$ .

${(4 {((4 + y) (4 - y))}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

Step 3.2.3.2

Simplify the left side.

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Step 3.2.3.2.1

Simplify ${(4 {((4 + y) (4 - y))}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.3.2.1.1

Expand $(4 + y) (4 - y)$ using the FOIL Method.

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Step 3.2.3.2.1.1.1

Apply the distributive property.

${(4 {(4 (4 - y) + y (4 - y))}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

Step 3.2.3.2.1.1.2

Apply the distributive property.

${(4 {(4 \cdot 4 + 4 (- y) + y (4 - y))}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

Step 3.2.3.2.1.1.3

Apply the distributive property.

${(4 {(4 \cdot 4 + 4 (- y) + y \cdot 4 + y (- y))}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

${(4 {(4 \cdot 4 + 4 (- y) + y \cdot 4 + y (- y))}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

Step 3.2.3.2.1.2

Simplify and combine like terms.

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Step 3.2.3.2.1.2.1

Simplify each term.

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Step 3.2.3.2.1.2.1.1

Multiply $4$ by $4$ .

${(4 {(16 + 4 (- y) + y \cdot 4 + y (- y))}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

Step 3.2.3.2.1.2.1.2

Multiply $- 1$ by $4$ .

${(4 {(16 - 4 y + y \cdot 4 + y (- y))}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

Step 3.2.3.2.1.2.1.3

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

${(4 {(16 - 4 y + 4 \cdot y + y (- y))}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

Step 3.2.3.2.1.2.1.4

Rewrite using the commutative property of multiplication.

${(4 {(16 - 4 y + 4 y - y \cdot y)}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

Step 3.2.3.2.1.2.1.5

Multiply $y$ by $y$ by adding the exponents.

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Step 3.2.3.2.1.2.1.5.1

Move $y$ .

${(4 {(16 - 4 y + 4 y - (y \cdot y))}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

Step 3.2.3.2.1.2.1.5.2

Multiply $y$ by $y$ .

${(4 {(16 - 4 y + 4 y - y^{2})}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

${(4 {(16 - 4 y + 4 y - y^{2})}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

${(4 {(16 - 4 y + 4 y - y^{2})}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

Step 3.2.3.2.1.2.2

Add $- 4 y$ and $4 y$ .

${(4 {(16 + 0 - y^{2})}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

Step 3.2.3.2.1.2.3

Add $16$ and $0$ .

${(4 {(16 - y^{2})}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

${(4 {(16 - y^{2})}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

Step 3.2.3.2.1.3

Apply the product rule to $4 {(16 - y^{2})}^{\frac{1}{2}}$ .

$4^{2} {({(16 - y^{2})}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

Step 3.2.3.2.1.4

Raise $4$ to the power of $2$ .

$16 {({(16 - y^{2})}^{\frac{1}{2}})}^{2} = {(- 4)}^{2}$

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

Step 3.2.3.2.1.5

Multiply the exponents in ${({(16 - y^{2})}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.3.2.1.5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$16 {(16 - y^{2})}^{\frac{1}{2} \cdot 2} = {(- 4)}^{2}$

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

Step 3.2.3.2.1.5.2

Cancel the common factor of $2$ .

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Step 3.2.3.2.1.5.2.1

Cancel the common factor.

$16 {(16 - y^{2})}^{\frac{1}{2} \cdot 2} = {(- 4)}^{2}$

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

Step 3.2.3.2.1.5.2.2

Rewrite the expression.

$16 (16 - y^{2}) = {(- 4)}^{2}$

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

$16 (16 - y^{2}) = {(- 4)}^{2}$

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

$16 (16 - y^{2}) = {(- 4)}^{2}$

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

Step 3.2.3.2.1.6

Simplify.

$16 (16 - y^{2}) = {(- 4)}^{2}$

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

Step 3.2.3.2.1.7

Apply the distributive property.

$16 \cdot 16 + 16 (- y^{2}) = {(- 4)}^{2}$

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

Step 3.2.3.2.1.8

Multiply.

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Step 3.2.3.2.1.8.1

Multiply $16$ by $16$ .

$256 + 16 (- y^{2}) = {(- 4)}^{2}$

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

Step 3.2.3.2.1.8.2

Multiply $- 1$ by $16$ .

$256 - 16 y^{2} = {(- 4)}^{2}$

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

$256 - 16 y^{2} = {(- 4)}^{2}$

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

$256 - 16 y^{2} = {(- 4)}^{2}$

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

$256 - 16 y^{2} = {(- 4)}^{2}$

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

Step 3.2.3.3

Simplify the right side.

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Step 3.2.3.3.1

Raise $- 4$ to the power of $2$ .

$256 - 16 y^{2} = 16$

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

$256 - 16 y^{2} = 16$

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

$256 - 16 y^{2} = 16$

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

Step 3.2.4

Solve for $y$ .

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Step 3.2.4.1

Move all terms not containing $y$ to the right side of the equation.

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Step 3.2.4.1.1

Subtract $256$ from both sides of the equation.

$- 16 y^{2} = 16 - 256$

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

Step 3.2.4.1.2

Subtract $256$ from $16$ .

$- 16 y^{2} = - 240$

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

$- 16 y^{2} = - 240$

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

Step 3.2.4.2

Divide each term in $- 16 y^{2} = - 240$ by $- 16$ and simplify.

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Step 3.2.4.2.1

Divide each term in $- 16 y^{2} = - 240$ by $- 16$ .

$\frac{- 16 y^{2}}{- 16} = \frac{- 240}{- 16}$

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

Step 3.2.4.2.2

Simplify the left side.

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Step 3.2.4.2.2.1

Cancel the common factor of $- 16$ .

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Step 3.2.4.2.2.1.1

Cancel the common factor.

$\frac{- 16 y^{2}}{- 16} = \frac{- 240}{- 16}$

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

Step 3.2.4.2.2.1.2

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{- 240}{- 16}$

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

$y^{2} = \frac{- 240}{- 16}$

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

$y^{2} = \frac{- 240}{- 16}$

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

Step 3.2.4.2.3

Simplify the right side.

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Step 3.2.4.2.3.1

Divide $- 240$ by $- 16$ .

$y^{2} = 15$

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

$y^{2} = 15$

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

$y^{2} = 15$

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

Step 3.2.4.3

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

$y = \pm \sqrt{15}$

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

Step 3.2.4.4

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

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Step 3.2.4.4.1

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

$y = \sqrt{15}$

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

Step 3.2.4.4.2

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

$y = - \sqrt{15}$

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

Step 3.2.4.4.3

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

$y = \sqrt{15}, - \sqrt{15}$

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

$y = \sqrt{15}, - \sqrt{15}$

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

$y = \sqrt{15}, - \sqrt{15}$

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

Step 3.2.5

Exclude the solutions that do not make $16 + 4 \sqrt{(4 + y) (4 - y)} = 12$ true.

No solution

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

No solution

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

No solution

Step 4