Solve Using the Square Root Property (n-7)^2=2n

${(n - 7)}^{2} = 2 n$

Step 1

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

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

${(n - 7)}^{2} - 2 n = 0$

Simplify each term.

Rewrite ${(n - 7)}^{2}$ as $(n - 7) (n - 7)$ .

$(n - 7) (n - 7) - 2 n = 0$

Expand $(n - 7) (n - 7)$ using the FOIL Method.

Apply the distributive property.

$n (n - 7) - 7 (n - 7) - 2 n = 0$

Apply the distributive property.

$n \cdot n + n \cdot - 7 - 7 (n - 7) - 2 n = 0$

Apply the distributive property.

$n \cdot n + n \cdot - 7 - 7 n - 7 \cdot - 7 - 2 n = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $n$ by $n$ .

$n^{2} + n \cdot - 7 - 7 n - 7 \cdot - 7 - 2 n = 0$

Move $- 7$ to the left of $n$ .

$n^{2} - 7 \cdot n - 7 n - 7 \cdot - 7 - 2 n = 0$

Multiply $- 7$ by $- 7$ .

$n^{2} - 7 n - 7 n + 49 - 2 n = 0$

Subtract $7 n$ from $- 7 n$ .

$n^{2} - 14 n + 49 - 2 n = 0$

Subtract $2 n$ from $- 14 n$ .

$n^{2} - 16 n + 49 = 0$

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 $a = 1$ , $b = - 16$ , and $c = 49$ into the quadratic formula and solve for $n$ .

$\frac{16 \pm \sqrt{{(- 16)}^{2} - 4 \cdot (1 \cdot 49)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 16$ to the power of $2$ .

$n = \frac{16 \pm \sqrt{256 - 4 \cdot 1 \cdot 49}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 49$ .

Multiply $- 4$ by $1$ .

$n = \frac{16 \pm \sqrt{256 - 4 \cdot 49}}{2 \cdot 1}$

Multiply $- 4$ by $49$ .

$n = \frac{16 \pm \sqrt{256 - 196}}{2 \cdot 1}$

Subtract $196$ from $256$ .

$n = \frac{16 \pm \sqrt{60}}{2 \cdot 1}$

Rewrite $60$ as $2^{2} \cdot 15$ .

Factor $4$ out of $60$ .

$n = \frac{16 \pm \sqrt{4 (15)}}{2 \cdot 1}$

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

$n = \frac{16 \pm \sqrt{2^{2} \cdot 15}}{2 \cdot 1}$

Pull terms out from under the radical.

$n = \frac{16 \pm 2 \sqrt{15}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$n = \frac{16 \pm 2 \sqrt{15}}{2}$

Simplify $\frac{16 \pm 2 \sqrt{15}}{2}$ .

$n = 8 \pm \sqrt{15}$

Step 5

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

Simplify the numerator.

Raise $- 16$ to the power of $2$ .

$n = \frac{16 \pm \sqrt{256 - 4 \cdot 1 \cdot 49}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 49$ .

Multiply $- 4$ by $1$ .

$n = \frac{16 \pm \sqrt{256 - 4 \cdot 49}}{2 \cdot 1}$

Multiply $- 4$ by $49$ .

$n = \frac{16 \pm \sqrt{256 - 196}}{2 \cdot 1}$

Subtract $196$ from $256$ .

$n = \frac{16 \pm \sqrt{60}}{2 \cdot 1}$

Rewrite $60$ as $2^{2} \cdot 15$ .

Factor $4$ out of $60$ .

$n = \frac{16 \pm \sqrt{4 (15)}}{2 \cdot 1}$

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

$n = \frac{16 \pm \sqrt{2^{2} \cdot 15}}{2 \cdot 1}$

Pull terms out from under the radical.

$n = \frac{16 \pm 2 \sqrt{15}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$n = \frac{16 \pm 2 \sqrt{15}}{2}$

Simplify $\frac{16 \pm 2 \sqrt{15}}{2}$ .

$n = 8 \pm \sqrt{15}$

Change the $\pm$ to $+$ .

$n = 8 + \sqrt{15}$

Step 6

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

Simplify the numerator.

Raise $- 16$ to the power of $2$ .

$n = \frac{16 \pm \sqrt{256 - 4 \cdot 1 \cdot 49}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 49$ .

Multiply $- 4$ by $1$ .

$n = \frac{16 \pm \sqrt{256 - 4 \cdot 49}}{2 \cdot 1}$

Multiply $- 4$ by $49$ .

$n = \frac{16 \pm \sqrt{256 - 196}}{2 \cdot 1}$

Subtract $196$ from $256$ .

$n = \frac{16 \pm \sqrt{60}}{2 \cdot 1}$

Rewrite $60$ as $2^{2} \cdot 15$ .

Factor $4$ out of $60$ .

$n = \frac{16 \pm \sqrt{4 (15)}}{2 \cdot 1}$

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

$n = \frac{16 \pm \sqrt{2^{2} \cdot 15}}{2 \cdot 1}$

Pull terms out from under the radical.

$n = \frac{16 \pm 2 \sqrt{15}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$n = \frac{16 \pm 2 \sqrt{15}}{2}$

Simplify $\frac{16 \pm 2 \sqrt{15}}{2}$ .

$n = 8 \pm \sqrt{15}$

Change the $\pm$ to $-$ .

$n = 8 - \sqrt{15}$

Step 7

The final answer is the combination of both solutions.

$n = 8 + \sqrt{15}, 8 - \sqrt{15}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$n = 8 + \sqrt{15}, 8 - \sqrt{15}$

Decimal Form:

$n = 11.87298334 \dots, 4.12701665 \dots$

Solve Using the Square Root Property (n-7)^2=2n

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