Solve 9=n-4/3n^2+2/3n^2

$$9 = \frac{ n-4 }{ 3 { n }^{ 2 } } + \frac{ 2 }{ 3 { n }^{ 2 } }$$

Solve for n

$$n=\frac{1+\sqrt{215}i}{54}\approx 0.018518519+0.271534783i$$

$$n=\frac{-\sqrt{215}i+1}{54}\approx 0.018518519-0.271534783i$$

Steps Using the Quadratic Formula

Variable n cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 3n^{2}.

$$27n^{2}=n-4+2$$

Add -4 and 2 to get -2.

$$27n^{2}=n-2$$

Subtract n from both sides.

$$27n^{2}-n=-2$$

Add 2 to both sides.

$$27n^{2}-n+2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 27 for a, -1 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-1\right)±\sqrt{1-4\times 27\times 2}}{2\times 27}$$

Multiply -4 times 27.

$$n=\frac{-\left(-1\right)±\sqrt{1-108\times 2}}{2\times 27}$$

Multiply -108 times 2.

$$n=\frac{-\left(-1\right)±\sqrt{1-216}}{2\times 27}$$

Add 1 to -216.

$$n=\frac{-\left(-1\right)±\sqrt{-215}}{2\times 27}$$

Take the square root of -215.

$$n=\frac{-\left(-1\right)±\sqrt{215}i}{2\times 27}$$

The opposite of -1 is 1.

$$n=\frac{1±\sqrt{215}i}{2\times 27}$$

Multiply 2 times 27.

$$n=\frac{1±\sqrt{215}i}{54}$$

Now solve the equation n=\frac{1±\sqrt{215}i}{54} when ± is plus. Add 1 to i\sqrt{215}.

$$n=\frac{1+\sqrt{215}i}{54}$$

Now solve the equation n=\frac{1±\sqrt{215}i}{54} when ± is minus. Subtract i\sqrt{215} from 1.

$$n=\frac{-\sqrt{215}i+1}{54}$$

The equation is now solved.

$$n=\frac{1+\sqrt{215}i}{54} n=\frac{-\sqrt{215}i+1}{54}$$

Steps for Completing the Square

Variable n cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 3n^{2}.

$$27n^{2}=n-4+2$$

Add -4 and 2 to get -2.

$$27n^{2}=n-2$$

Subtract n from both sides.

$$27n^{2}-n=-2$$

Divide both sides by 27.

$$\frac{27n^{2}-n}{27}=-\frac{2}{27}$$

Dividing by 27 undoes the multiplication by 27.

$$n^{2}-\frac{1}{27}n=-\frac{2}{27}$$

Divide -\frac{1}{27}, the coefficient of the x term, by 2 to get -\frac{1}{54}. Then add the square of -\frac{1}{54} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-\frac{1}{27}n+\left(-\frac{1}{54}\right)^{2}=-\frac{2}{27}+\left(-\frac{1}{54}\right)^{2}$$

Square -\frac{1}{54} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-\frac{1}{27}n+\frac{1}{2916}=-\frac{2}{27}+\frac{1}{2916}$$

Add -\frac{2}{27} to \frac{1}{2916} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$n^{2}-\frac{1}{27}n+\frac{1}{2916}=-\frac{215}{2916}$$

Factor n^{2}-\frac{1}{27}n+\frac{1}{2916}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-\frac{1}{54}\right)^{2}=-\frac{215}{2916}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{1}{54}\right)^{2}}=\sqrt{-\frac{215}{2916}}$$

Simplify.

$$n-\frac{1}{54}=\frac{\sqrt{215}i}{54} n-\frac{1}{54}=-\frac{\sqrt{215}i}{54}$$

Add \frac{1}{54} to both sides of the equation.

$$n=\frac{1+\sqrt{215}i}{54} n=\frac{-\sqrt{215}i+1}{54}$$