Solve n^2-2n=35^2

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

Solve for n

$$n=\sqrt{1226}+1\approx 36.0142828$$

$$n=1-\sqrt{1226}\approx -34.0142828$$

Steps Using the Quadratic Formula

Calculate 35 to the power of 2 and get 1225.

$$n^{2}-2n=1225$$

Subtract 1225 from both sides.

$$n^{2}-2n-1225=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and -1225 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-1225\right)}}{2}$$

Square -2.

$$n=\frac{-\left(-2\right)±\sqrt{4-4\left(-1225\right)}}{2}$$

Multiply -4 times -1225.

$$n=\frac{-\left(-2\right)±\sqrt{4+4900}}{2}$$

Add 4 to 4900.

$$n=\frac{-\left(-2\right)±\sqrt{4904}}{2}$$

Take the square root of 4904.

$$n=\frac{-\left(-2\right)±2\sqrt{1226}}{2}$$

The opposite of -2 is 2.

$$n=\frac{2±2\sqrt{1226}}{2}$$

Now solve the equation n=\frac{2±2\sqrt{1226}}{2} when ± is plus. Add 2 to 2\sqrt{1226}.

$$n=\frac{2\sqrt{1226}+2}{2}$$

Divide 2+2\sqrt{1226} by 2.

$$n=\sqrt{1226}+1$$

Now solve the equation n=\frac{2±2\sqrt{1226}}{2} when ± is minus. Subtract 2\sqrt{1226} from 2.

$$n=\frac{2-2\sqrt{1226}}{2}$$

Divide 2-2\sqrt{1226} by 2.

$$n=1-\sqrt{1226}$$

The equation is now solved.

$$n=\sqrt{1226}+1 n=1-\sqrt{1226}$$

Steps for Completing the Square

Calculate 35 to the power of 2 and get 1225.

$$n^{2}-2n=1225$$

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.

$$n^{2}-2n+1=1225+1$$

Add 1225 to 1.

$$n^{2}-2n+1=1226$$

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}.

$$\left(n-1\right)^{2}=1226$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-1\right)^{2}}=\sqrt{1226}$$

Simplify.

$$n-1=\sqrt{1226} n-1=-\sqrt{1226}$$

Add 1 to both sides of the equation.

$$n=\sqrt{1226}+1 n=1-\sqrt{1226}$$