Solve -4=n(2(9)(n-1)-2)

$$- 4 = n ( 2 ( 9 ) ( n - 1 ) - 2 )$$

Solve for n

$$n=\frac{\sqrt{7}+5}{9}\approx 0.849527923$$

$$n=\frac{5-\sqrt{7}}{9}\approx 0.261583188$$

Steps Using the Quadratic Formula

Multiply 2 and 9 to get 18.

$$-4=n\left(18\left(n-1\right)-2\right)$$

Use the distributive property to multiply 18 by n-1.

$$-4=n\left(18n-18-2\right)$$

Subtract 2 from -18 to get -20.

$$-4=n\left(18n-20\right)$$

Use the distributive property to multiply n by 18n-20.

$$-4=18n^{2}-20n$$

Swap sides so that all variable terms are on the left hand side.

$$18n^{2}-20n=-4$$

Add 4 to both sides.

$$18n^{2}-20n+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 18 for a, -20 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\times 18\times 4}}{2\times 18}$$

Square -20.

$$n=\frac{-\left(-20\right)±\sqrt{400-4\times 18\times 4}}{2\times 18}$$

Multiply -4 times 18.

$$n=\frac{-\left(-20\right)±\sqrt{400-72\times 4}}{2\times 18}$$

Multiply -72 times 4.

$$n=\frac{-\left(-20\right)±\sqrt{400-288}}{2\times 18}$$

Add 400 to -288.

$$n=\frac{-\left(-20\right)±\sqrt{112}}{2\times 18}$$

Take the square root of 112.

$$n=\frac{-\left(-20\right)±4\sqrt{7}}{2\times 18}$$

The opposite of -20 is 20.

$$n=\frac{20±4\sqrt{7}}{2\times 18}$$

Multiply 2 times 18.

$$n=\frac{20±4\sqrt{7}}{36}$$

Now solve the equation n=\frac{20±4\sqrt{7}}{36} when ± is plus. Add 20 to 4\sqrt{7}.

$$n=\frac{4\sqrt{7}+20}{36}$$

Divide 20+4\sqrt{7} by 36.

$$n=\frac{\sqrt{7}+5}{9}$$

Now solve the equation n=\frac{20±4\sqrt{7}}{36} when ± is minus. Subtract 4\sqrt{7} from 20.

$$n=\frac{20-4\sqrt{7}}{36}$$

Divide 20-4\sqrt{7} by 36.

$$n=\frac{5-\sqrt{7}}{9}$$

The equation is now solved.

$$n=\frac{\sqrt{7}+5}{9} n=\frac{5-\sqrt{7}}{9}$$

Steps for Completing the Square

Multiply 2 and 9 to get 18.

$$-4=n\left(18\left(n-1\right)-2\right)$$

Use the distributive property to multiply 18 by n-1.

$$-4=n\left(18n-18-2\right)$$

Subtract 2 from -18 to get -20.

$$-4=n\left(18n-20\right)$$

Use the distributive property to multiply n by 18n-20.

$$-4=18n^{2}-20n$$

Swap sides so that all variable terms are on the left hand side.

$$18n^{2}-20n=-4$$

Divide both sides by 18.

$$\frac{18n^{2}-20n}{18}=-\frac{4}{18}$$

Dividing by 18 undoes the multiplication by 18.

$$n^{2}+\left(-\frac{20}{18}\right)n=-\frac{4}{18}$$

Reduce the fraction \frac{-20}{18} to lowest terms by extracting and canceling out 2.

$$n^{2}-\frac{10}{9}n=-\frac{4}{18}$$

Reduce the fraction \frac{-4}{18} to lowest terms by extracting and canceling out 2.

$$n^{2}-\frac{10}{9}n=-\frac{2}{9}$$

Divide -\frac{10}{9}, the coefficient of the x term, by 2 to get -\frac{5}{9}. Then add the square of -\frac{5}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-\frac{10}{9}n+\left(-\frac{5}{9}\right)^{2}=-\frac{2}{9}+\left(-\frac{5}{9}\right)^{2}$$

Square -\frac{5}{9} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-\frac{10}{9}n+\frac{25}{81}=-\frac{2}{9}+\frac{25}{81}$$

Add -\frac{2}{9} to \frac{25}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$n^{2}-\frac{10}{9}n+\frac{25}{81}=\frac{7}{81}$$

Factor n^{2}-\frac{10}{9}n+\frac{25}{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}.

$$\left(n-\frac{5}{9}\right)^{2}=\frac{7}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{5}{9}\right)^{2}}=\sqrt{\frac{7}{81}}$$

Simplify.

$$n-\frac{5}{9}=\frac{\sqrt{7}}{9} n-\frac{5}{9}=-\frac{\sqrt{7}}{9}$$

Add \frac{5}{9} to both sides of the equation.

$$n=\frac{\sqrt{7}+5}{9} n=\frac{5-\sqrt{7}}{9}$$