Solve for a n+2 = square root of a-n

$n + 2 = \sqrt{a - n}$

Rewrite the equation as $\sqrt{a - n} = n + 2$ .

$\sqrt{a - n} = n + 2$

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

${\sqrt{a - n}}^{2} = {(n + 2)}^{2}$

Simplify each side of the equation.

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

${({(a - n)}^{\frac{1}{2}})}^{2} = {(n + 2)}^{2}$

Simplify the left side.

Simplify ${({(a - n)}^{\frac{1}{2}})}^{2}$ .

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

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

${(a - n)}^{\frac{1}{2} \cdot 2} = {(n + 2)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(a - n)}^{\frac{1}{2} \cdot 2} = {(n + 2)}^{2}$

Rewrite the expression.

${(a - n)}^{1} = {(n + 2)}^{2}$

Simplify.

$a - n = {(n + 2)}^{2}$

Simplify the right side.

Simplify ${(n + 2)}^{2}$ .

Rewrite ${(n + 2)}^{2}$ as $(n + 2) (n + 2)$ .

$a - n = (n + 2) (n + 2)$

Expand $(n + 2) (n + 2)$ using the FOIL Method.

Apply the distributive property.

$a - n = n (n + 2) + 2 (n + 2)$

Apply the distributive property.

$a - n = n \cdot n + n \cdot 2 + 2 (n + 2)$

Apply the distributive property.

$a - n = n \cdot n + n \cdot 2 + 2 n + 2 \cdot 2$

Simplify and combine like terms.

Simplify each term.

Multiply $n$ by $n$ .

$a - n = n^{2} + n \cdot 2 + 2 n + 2 \cdot 2$

Move $2$ to the left of $n$ .

$a - n = n^{2} + 2 \cdot n + 2 n + 2 \cdot 2$

Multiply $2$ by $2$ .

$a - n = n^{2} + 2 n + 2 n + 4$

Add $2 n$ and $2 n$ .

$a - n = n^{2} + 4 n + 4$

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

Add $n$ to both sides of the equation.

$a = n^{2} + 4 n + 4 + n$

Add $4 n$ and $n$ .

$a = n^{2} + 5 n + 4$

Solve for a n+2 = square root of a-n

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