Solve for n 20=n+ square root of n

$20 = n + \sqrt{n}$

Rewrite the equation as $n + \sqrt{n} = 20$ .

$n + \sqrt{n} = 20$

Subtract $n$ from both sides of the equation.

$\sqrt{n} = 20 - n$

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

${\sqrt{n}}^{2} = {(20 - n)}^{2}$

Simplify each side of the equation.

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

${(n^{\frac{1}{2}})}^{2} = {(20 - n)}^{2}$

Simplify the left side.

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

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

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

$n^{\frac{1}{2} \cdot 2} = {(20 - n)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$n^{\frac{1}{2} \cdot 2} = {(20 - n)}^{2}$

Rewrite the expression.

$n^{1} = {(20 - n)}^{2}$

Simplify.

$n = {(20 - n)}^{2}$

Simplify the right side.

Simplify ${(20 - n)}^{2}$ .

Rewrite ${(20 - n)}^{2}$ as $(20 - n) (20 - n)$ .

$n = (20 - n) (20 - n)$

Expand $(20 - n) (20 - n)$ using the FOIL Method.

Apply the distributive property.

$n = 20 (20 - n) - n (20 - n)$

Apply the distributive property.

$n = 20 \cdot 20 + 20 (- n) - n (20 - n)$

Apply the distributive property.

$n = 20 \cdot 20 + 20 (- n) - n \cdot 20 - n (- n)$

Simplify and combine like terms.

Simplify each term.

Multiply $20$ by $20$ .

$n = 400 + 20 (- n) - n \cdot 20 - n (- n)$

Multiply $- 1$ by $20$ .

$n = 400 - 20 n - n \cdot 20 - n (- n)$

Multiply $20$ by $- 1$ .

$n = 400 - 20 n - 20 n - n (- n)$

Rewrite using the commutative property of multiplication.

$n = 400 - 20 n - 20 n - 1 \cdot - 1 n \cdot n$

Multiply $n$ by $n$ by adding the exponents.

Move $n$ .

$n = 400 - 20 n - 20 n - 1 \cdot - 1 (n \cdot n)$

Multiply $n$ by $n$ .

$n = 400 - 20 n - 20 n - 1 \cdot - 1 n^{2}$

Multiply $- 1$ by $- 1$ .

$n = 400 - 20 n - 20 n + 1 n^{2}$

Multiply $n^{2}$ by $1$ .

$n = 400 - 20 n - 20 n + n^{2}$

Subtract $20 n$ from $- 20 n$ .

$n = 400 - 40 n + n^{2}$

Solve for $n$ .

Since $n$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$400 - 40 n + n^{2} = n$

Move all terms containing $n$ to the left side of the equation.

Subtract $n$ from both sides of the equation.

$400 - 40 n + n^{2} - n = 0$

Subtract $n$ from $- 40 n$ .

$400 + n^{2} - 41 n = 0$

Factor $400 + n^{2} - 41 n$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $400$ and whose sum is $- 41$ .

$- 25, - 16$

Write the factored form using these integers.

$(n - 25) (n - 16) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$n - 25 = 0$

$n - 16 = 0$

Set $n - 25$ equal to $0$ and solve for $n$ .

Set $n - 25$ equal to $0$ .

$n - 25 = 0$

Add $25$ to both sides of the equation.

$n = 25$

Set $n - 16$ equal to $0$ and solve for $n$ .

Set $n - 16$ equal to $0$ .

$n - 16 = 0$

Add $16$ to both sides of the equation.

$n = 16$

The final solution is all the values that make $(n - 25) (n - 16) = 0$ true.

$n = 25, 16$

Exclude the solutions that do not make $20 = n + \sqrt{n}$ true.

$n = 16$

Solve for n 20=n+ square root of n

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