Solve for n square root of -5+6n=n

$\sqrt{- 5 + 6 n} = n$

Step 1

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

${\sqrt{- 5 + 6 n}}^{2} = n^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{- 5 + 6 n}$ as ${(- 5 + 6 n)}^{\frac{1}{2}}$ .

${({(- 5 + 6 n)}^{\frac{1}{2}})}^{2} = n^{2}$

Simplify the left side.

Simplify ${({(- 5 + 6 n)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(- 5 + 6 n)}^{\frac{1}{2}})}^{2}$ .

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

${(- 5 + 6 n)}^{\frac{1}{2} \cdot 2} = n^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(- 5 + 6 n)}^{\frac{1}{2} \cdot 2} = n^{2}$

Rewrite the expression.

${(- 5 + 6 n)}^{1} = n^{2}$

Simplify.

$- 5 + 6 n = n^{2}$

Step 3

Solve for $n$ .

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

$- 5 + 6 n - n^{2} = 0$

Factor the left side of the equation.

Factor $- 1$ out of $- 5 + 6 n - n^{2}$ .

Reorder the expression.

Move $- 5$ .

$6 n - n^{2} - 5 = 0$

Reorder $6 n$ and $- n^{2}$ .

$- n^{2} + 6 n - 5 = 0$

Factor $- 1$ out of $- n^{2}$ .

$- (n^{2}) + 6 n - 5 = 0$

Factor $- 1$ out of $6 n$ .

$- (n^{2}) - (- 6 n) - 5 = 0$

Rewrite $- 5$ as $- 1 (5)$ .

$- (n^{2}) - (- 6 n) - 1 \cdot 5 = 0$

Factor $- 1$ out of $- (n^{2}) - (- 6 n)$ .

$- (n^{2} - 6 n) - 1 \cdot 5 = 0$

Factor $- 1$ out of $- (n^{2} - 6 n) - 1 (5)$ .

$- (n^{2} - 6 n + 5) = 0$

Factor.

Factor $n^{2} - 6 n + 5$ 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 $5$ and whose sum is $- 6$ .

$- 5, - 1$

Write the factored form using these integers.

$- ((n - 5) (n - 1)) = 0$

Remove unnecessary parentheses.

$- (n - 5) (n - 1) = 0$

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

$n - 5 = 0$

$n - 1 = 0$

Set $n - 5$ equal to $0$ and solve for $n$ .

Set $n - 5$ equal to $0$ .

$n - 5 = 0$

Add $5$ to both sides of the equation.

$n = 5$

Set $n - 1$ equal to $0$ and solve for $n$ .

Set $n - 1$ equal to $0$ .

$n - 1 = 0$

Add $1$ to both sides of the equation.

$n = 1$

The final solution is all the values that make $- (n - 5) (n - 1) = 0$ true.

$n = 5, 1$

Solve for n square root of -5+6n=n

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