Solve n^2=18n+4

$$n ^ { 2 } = 18 n + 4$$

Solve for n

$$n=\sqrt{85}+9\approx 18.219544457$$

$$n=9-\sqrt{85}\approx -0.219544457$$

Steps Using the Quadratic Formula

Subtract 18n from both sides.

$$n^{2}-18n=4$$

Subtract 4 from both sides.

$$n^{2}-18n-4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -18 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\left(-4\right)}}{2}$$

Square -18.

$$n=\frac{-\left(-18\right)±\sqrt{324-4\left(-4\right)}}{2}$$

Multiply -4 times -4.

$$n=\frac{-\left(-18\right)±\sqrt{324+16}}{2}$$

Add 324 to 16.

$$n=\frac{-\left(-18\right)±\sqrt{340}}{2}$$

Take the square root of 340.

$$n=\frac{-\left(-18\right)±2\sqrt{85}}{2}$$

The opposite of -18 is 18.

$$n=\frac{18±2\sqrt{85}}{2}$$

Now solve the equation n=\frac{18±2\sqrt{85}}{2} when ± is plus. Add 18 to 2\sqrt{85}.

$$n=\frac{2\sqrt{85}+18}{2}$$

Divide 18+2\sqrt{85} by 2.

$$n=\sqrt{85}+9$$

Now solve the equation n=\frac{18±2\sqrt{85}}{2} when ± is minus. Subtract 2\sqrt{85} from 18.

$$n=\frac{18-2\sqrt{85}}{2}$$

Divide 18-2\sqrt{85} by 2.

$$n=9-\sqrt{85}$$

The equation is now solved.

$$n=\sqrt{85}+9 n=9-\sqrt{85}$$

Steps for Completing the Square

Subtract 18n from both sides.

$$n^{2}-18n=4$$

Divide -18, the coefficient of the x term, by 2 to get -9. Then add the square of -9 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-18n+\left(-9\right)^{2}=4+\left(-9\right)^{2}$$

Square -9.

$$n^{2}-18n+81=4+81$$

Add 4 to 81.

$$n^{2}-18n+81=85$$

Factor n^{2}-18n+81. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-9\right)^{2}=85$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-9\right)^{2}}=\sqrt{85}$$

Simplify.

$$n-9=\sqrt{85} n-9=-\sqrt{85}$$

Add 9 to both sides of the equation.

$$n=\sqrt{85}+9 n=9-\sqrt{85}$$