Solve 3n/n-2=2/n-1

$$\frac { 3 n } { n - 2 } = \frac { 2 } { n - 1 }$$

Solve for n

$$n=\frac{5+\sqrt{23}i}{6}\approx 0.833333333+0.799305254i$$

$$n=\frac{-\sqrt{23}i+5}{6}\approx 0.833333333-0.799305254i$$

Steps Using the Quadratic Formula

Variable n cannot be equal to any of the values 1,2 since division by zero is not defined. Multiply both sides of the equation by \left(n-2\right)\left(n-1\right), the least common multiple of n-2,n-1.

$$\left(n-1\right)\times 3n=\left(n-2\right)\times 2$$

Use the distributive property to multiply n-1 by 3.

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

Use the distributive property to multiply 3n-3 by n.

$$3n^{2}-3n=\left(n-2\right)\times 2$$

Use the distributive property to multiply n-2 by 2.

$$3n^{2}-3n=2n-4$$

Subtract 2n from both sides.

$$3n^{2}-3n-2n=-4$$

Combine -3n and -2n to get -5n.

$$3n^{2}-5n=-4$$

Add 4 to both sides.

$$3n^{2}-5n+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -5 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 3\times 4}}{2\times 3}$$

Square -5.

$$n=\frac{-\left(-5\right)±\sqrt{25-4\times 3\times 4}}{2\times 3}$$

Multiply -4 times 3.

$$n=\frac{-\left(-5\right)±\sqrt{25-12\times 4}}{2\times 3}$$

Multiply -12 times 4.

$$n=\frac{-\left(-5\right)±\sqrt{25-48}}{2\times 3}$$

Add 25 to -48.

$$n=\frac{-\left(-5\right)±\sqrt{-23}}{2\times 3}$$

Take the square root of -23.

$$n=\frac{-\left(-5\right)±\sqrt{23}i}{2\times 3}$$

The opposite of -5 is 5.

$$n=\frac{5±\sqrt{23}i}{2\times 3}$$

Multiply 2 times 3.

$$n=\frac{5±\sqrt{23}i}{6}$$

Now solve the equation n=\frac{5±\sqrt{23}i}{6} when ± is plus. Add 5 to i\sqrt{23}.

$$n=\frac{5+\sqrt{23}i}{6}$$

Now solve the equation n=\frac{5±\sqrt{23}i}{6} when ± is minus. Subtract i\sqrt{23} from 5.

$$n=\frac{-\sqrt{23}i+5}{6}$$

The equation is now solved.

$$n=\frac{5+\sqrt{23}i}{6} n=\frac{-\sqrt{23}i+5}{6}$$

Steps for Completing the Square

Variable n cannot be equal to any of the values 1,2 since division by zero is not defined. Multiply both sides of the equation by \left(n-2\right)\left(n-1\right), the least common multiple of n-2,n-1.

$$\left(n-1\right)\times 3n=\left(n-2\right)\times 2$$

Use the distributive property to multiply n-1 by 3.

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

Use the distributive property to multiply 3n-3 by n.

$$3n^{2}-3n=\left(n-2\right)\times 2$$

Use the distributive property to multiply n-2 by 2.

$$3n^{2}-3n=2n-4$$

Subtract 2n from both sides.

$$3n^{2}-3n-2n=-4$$

Combine -3n and -2n to get -5n.

$$3n^{2}-5n=-4$$

Divide both sides by 3.

$$\frac{3n^{2}-5n}{3}=-\frac{4}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$n^{2}-\frac{5}{3}n=-\frac{4}{3}$$

Divide -\frac{5}{3}, the coefficient of the x term, by 2 to get -\frac{5}{6}. Then add the square of -\frac{5}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-\frac{5}{3}n+\left(-\frac{5}{6}\right)^{2}=-\frac{4}{3}+\left(-\frac{5}{6}\right)^{2}$$

Square -\frac{5}{6} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-\frac{5}{3}n+\frac{25}{36}=-\frac{4}{3}+\frac{25}{36}$$

Add -\frac{4}{3} to \frac{25}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$n^{2}-\frac{5}{3}n+\frac{25}{36}=-\frac{23}{36}$$

Factor n^{2}-\frac{5}{3}n+\frac{25}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-\frac{5}{6}\right)^{2}=-\frac{23}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{5}{6}\right)^{2}}=\sqrt{-\frac{23}{36}}$$

Simplify.

$$n-\frac{5}{6}=\frac{\sqrt{23}i}{6} n-\frac{5}{6}=-\frac{\sqrt{23}i}{6}$$

Add \frac{5}{6} to both sides of the equation.

$$n=\frac{5+\sqrt{23}i}{6} n=\frac{-\sqrt{23}i+5}{6}$$