Solve n^2=2n-6^2

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

Solve for n

$$n=1+\sqrt{35}i\approx 1+5.916079783i$$

$$n=-\sqrt{35}i+1\approx 1-5.916079783i$$

Steps Using the Quadratic Formula

Calculate 6 to the power of 2 and get 36.

$$n^{2}=2n-36$$

Subtract 2n from both sides.

$$n^{2}-2n=-36$$

Add 36 to both sides.

$$n^{2}-2n+36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and 36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 36}}{2}$$

Square -2.

$$n=\frac{-\left(-2\right)±\sqrt{4-4\times 36}}{2}$$

Multiply -4 times 36.

$$n=\frac{-\left(-2\right)±\sqrt{4-144}}{2}$$

Add 4 to -144.

$$n=\frac{-\left(-2\right)±\sqrt{-140}}{2}$$

Take the square root of -140.

$$n=\frac{-\left(-2\right)±2\sqrt{35}i}{2}$$

The opposite of -2 is 2.

$$n=\frac{2±2\sqrt{35}i}{2}$$

Now solve the equation n=\frac{2±2\sqrt{35}i}{2} when ± is plus. Add 2 to 2i\sqrt{35}.

$$n=\frac{2+2\sqrt{35}i}{2}$$

Divide 2+2i\sqrt{35} by 2.

$$n=1+\sqrt{35}i$$

Now solve the equation n=\frac{2±2\sqrt{35}i}{2} when ± is minus. Subtract 2i\sqrt{35} from 2.

$$n=\frac{-2\sqrt{35}i+2}{2}$$

Divide 2-2i\sqrt{35} by 2.

$$n=-\sqrt{35}i+1$$

The equation is now solved.

$$n=1+\sqrt{35}i n=-\sqrt{35}i+1$$

Steps for Completing the Square

Calculate 6 to the power of 2 and get 36.

$$n^{2}=2n-36$$

Subtract 2n from both sides.

$$n^{2}-2n=-36$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-2n+1=-36+1$$

Add -36 to 1.

$$n^{2}-2n+1=-35$$

Factor n^{2}-2n+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-1\right)^{2}=-35$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-1\right)^{2}}=\sqrt{-35}$$

Simplify.

$$n-1=\sqrt{35}i n-1=-\sqrt{35}i$$

Add 1 to both sides of the equation.

$$n=1+\sqrt{35}i n=-\sqrt{35}i+1$$