Solve Using the Square Root Property (n-2)^2=2n

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

Step 1

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

Subtract $2 n$ from both sides of the equation.

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

Simplify each term.

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

$(n - 2) (n - 2) - 2 n = 0$

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

Apply the distributive property.

$n (n - 2) - 2 (n - 2) - 2 n = 0$

Apply the distributive property.

$n \cdot n + n \cdot - 2 - 2 (n - 2) - 2 n = 0$

Apply the distributive property.

$n \cdot n + n \cdot - 2 - 2 n - 2 \cdot - 2 - 2 n = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $n$ by $n$ .

$n^{2} + n \cdot - 2 - 2 n - 2 \cdot - 2 - 2 n = 0$

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

$n^{2} - 2 \cdot n - 2 n - 2 \cdot - 2 - 2 n = 0$

Multiply $- 2$ by $- 2$ .

$n^{2} - 2 n - 2 n + 4 - 2 n = 0$

Subtract $2 n$ from $- 2 n$ .

$n^{2} - 4 n + 4 - 2 n = 0$

Subtract $2 n$ from $- 4 n$ .

$n^{2} - 6 n + 4 = 0$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = 1$ , $b = - 6$ , and $c = 4$ into the quadratic formula and solve for $n$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (1 \cdot 4)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 6$ to the power of $2$ .

$n = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 4$ .

Multiply $- 4$ by $1$ .

$n = \frac{6 \pm \sqrt{36 - 4 \cdot 4}}{2 \cdot 1}$

Multiply $- 4$ by $4$ .

$n = \frac{6 \pm \sqrt{36 - 16}}{2 \cdot 1}$

Subtract $16$ from $36$ .

$n = \frac{6 \pm \sqrt{20}}{2 \cdot 1}$

Rewrite $20$ as $2^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$n = \frac{6 \pm \sqrt{4 (5)}}{2 \cdot 1}$

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

$n = \frac{6 \pm \sqrt{2^{2} \cdot 5}}{2 \cdot 1}$

Pull terms out from under the radical.

$n = \frac{6 \pm 2 \sqrt{5}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$n = \frac{6 \pm 2 \sqrt{5}}{2}$

Simplify $\frac{6 \pm 2 \sqrt{5}}{2}$ .

$n = 3 \pm \sqrt{5}$

Step 5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 6$ to the power of $2$ .

$n = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 4$ .

Multiply $- 4$ by $1$ .

$n = \frac{6 \pm \sqrt{36 - 4 \cdot 4}}{2 \cdot 1}$

Multiply $- 4$ by $4$ .

$n = \frac{6 \pm \sqrt{36 - 16}}{2 \cdot 1}$

Subtract $16$ from $36$ .

$n = \frac{6 \pm \sqrt{20}}{2 \cdot 1}$

Rewrite $20$ as $2^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$n = \frac{6 \pm \sqrt{4 (5)}}{2 \cdot 1}$

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

$n = \frac{6 \pm \sqrt{2^{2} \cdot 5}}{2 \cdot 1}$

Pull terms out from under the radical.

$n = \frac{6 \pm 2 \sqrt{5}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$n = \frac{6 \pm 2 \sqrt{5}}{2}$

Simplify $\frac{6 \pm 2 \sqrt{5}}{2}$ .

$n = 3 \pm \sqrt{5}$

Change the $\pm$ to $+$ .

$n = 3 + \sqrt{5}$

Step 6

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $- 6$ to the power of $2$ .

$n = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 4$ .

Multiply $- 4$ by $1$ .

$n = \frac{6 \pm \sqrt{36 - 4 \cdot 4}}{2 \cdot 1}$

Multiply $- 4$ by $4$ .

$n = \frac{6 \pm \sqrt{36 - 16}}{2 \cdot 1}$

Subtract $16$ from $36$ .

$n = \frac{6 \pm \sqrt{20}}{2 \cdot 1}$

Rewrite $20$ as $2^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$n = \frac{6 \pm \sqrt{4 (5)}}{2 \cdot 1}$

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

$n = \frac{6 \pm \sqrt{2^{2} \cdot 5}}{2 \cdot 1}$

Pull terms out from under the radical.

$n = \frac{6 \pm 2 \sqrt{5}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$n = \frac{6 \pm 2 \sqrt{5}}{2}$

Simplify $\frac{6 \pm 2 \sqrt{5}}{2}$ .

$n = 3 \pm \sqrt{5}$

Change the $\pm$ to $-$ .

$n = 3 - \sqrt{5}$

Step 7

The final answer is the combination of both solutions.

$n = 3 + \sqrt{5}, 3 - \sqrt{5}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$n = 3 + \sqrt{5}, 3 - \sqrt{5}$

Decimal Form:

$n = 5.23606797 \dots, 0.76393202 \dots$

Solve Using the Square Root Property (n-2)^2=2n

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