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

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

Solve for n

$$n=\sqrt{10}+1\approx 4.16227766$$

$$n=1-\sqrt{10}\approx -2.16227766$$

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^{3}, the least common multiple of n^{3},3n^{2}.

$$3\times 3=n\left(n-4\right)+n\times 2$$

Multiply 3 and 3 to get 9.

$$9=n\left(n-4\right)+n\times 2$$

Use the distributive property to multiply n by n-4.

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

Combine -4n and n\times 2 to get -2n.

$$9=n^{2}-2n$$

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

$$n^{2}-2n=9$$

Subtract 9 from both sides.

$$n^{2}-2n-9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-9\right)}}{2}$$

Square -2.

$$n=\frac{-\left(-2\right)±\sqrt{4-4\left(-9\right)}}{2}$$

Multiply -4 times -9.

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

Add 4 to 36.

$$n=\frac{-\left(-2\right)±\sqrt{40}}{2}$$

Take the square root of 40.

$$n=\frac{-\left(-2\right)±2\sqrt{10}}{2}$$

The opposite of -2 is 2.

$$n=\frac{2±2\sqrt{10}}{2}$$

Now solve the equation n=\frac{2±2\sqrt{10}}{2} when ± is plus. Add 2 to 2\sqrt{10}.

$$n=\frac{2\sqrt{10}+2}{2}$$

Divide 2+2\sqrt{10} by 2.

$$n=\sqrt{10}+1$$

Now solve the equation n=\frac{2±2\sqrt{10}}{2} when ± is minus. Subtract 2\sqrt{10} from 2.

$$n=\frac{2-2\sqrt{10}}{2}$$

Divide 2-2\sqrt{10} by 2.

$$n=1-\sqrt{10}$$

The equation is now solved.

$$n=\sqrt{10}+1 n=1-\sqrt{10}$$

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^{3}, the least common multiple of n^{3},3n^{2}.

$$3\times 3=n\left(n-4\right)+n\times 2$$

Multiply 3 and 3 to get 9.

$$9=n\left(n-4\right)+n\times 2$$

Use the distributive property to multiply n by n-4.

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

Combine -4n and n\times 2 to get -2n.

$$9=n^{2}-2n$$

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

$$n^{2}-2n=9$$

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.

$$n^{2}-2n+1=9+1$$

Add 9 to 1.

$$n^{2}-2n+1=10$$

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}.

$$\left(n-1\right)^{2}=10$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-1\right)^{2}}=\sqrt{10}$$

Simplify.

$$n-1=\sqrt{10} n-1=-\sqrt{10}$$

Add 1 to both sides of the equation.

$$n=\sqrt{10}+1 n=1-\sqrt{10}$$