Solve (N+2)(N-1)/N=4/3

$$\frac { ( N + 2 ) ( N - 1 ) } { N } = \frac { 4 } { 3 }$$

Solve for N

$$N = \frac{\sqrt{73} + 1}{6} \approx 1.590667291$$

$$N=\frac{1-\sqrt{73}}{6}\approx -1.257333958$$

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, the least common multiple of N,3.

$$3\left(N+2\right)\left(N-1\right)=4N$$

Use the distributive property to multiply 3 by N+2.

$$\left(3N+6\right)\left(N-1\right)=4N$$

Use the distributive property to multiply 3N+6 by N-1 and combine like terms.

$$3N^{2}+3N-6=4N$$

Subtract 4N from both sides.

$$3N^{2}+3N-6-4N=0$$

Combine 3N and -4N to get -N.

$$3N^{2}-N-6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -1 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$N=\frac{-\left(-1\right)±\sqrt{1-4\times 3\left(-6\right)}}{2\times 3}$$

Multiply -4 times 3.

$$N=\frac{-\left(-1\right)±\sqrt{1-12\left(-6\right)}}{2\times 3}$$

Multiply -12 times -6.

$$N=\frac{-\left(-1\right)±\sqrt{1+72}}{2\times 3}$$

Add 1 to 72.

$$N=\frac{-\left(-1\right)±\sqrt{73}}{2\times 3}$$

The opposite of -1 is 1.

$$N=\frac{1±\sqrt{73}}{2\times 3}$$

Multiply 2 times 3.

$$N=\frac{1±\sqrt{73}}{6}$$

Now solve the equation N=\frac{1±\sqrt{73}}{6} when ± is plus. Add 1 to \sqrt{73}.

$$N=\frac{\sqrt{73}+1}{6}$$

Now solve the equation N=\frac{1±\sqrt{73}}{6} when ± is minus. Subtract \sqrt{73} from 1.

$$N=\frac{1-\sqrt{73}}{6}$$

The equation is now solved.

$$N=\frac{\sqrt{73}+1}{6} N=\frac{1-\sqrt{73}}{6}$$

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, the least common multiple of N,3.

$$3\left(N+2\right)\left(N-1\right)=4N$$

Use the distributive property to multiply 3 by N+2.

$$\left(3N+6\right)\left(N-1\right)=4N$$

Use the distributive property to multiply 3N+6 by N-1 and combine like terms.

$$3N^{2}+3N-6=4N$$

Subtract 4N from both sides.

$$3N^{2}+3N-6-4N=0$$

Combine 3N and -4N to get -N.

$$3N^{2}-N-6=0$$

Add 6 to both sides. Anything plus zero gives itself.

$$3N^{2}-N=6$$

Divide both sides by 3.

$$\frac{3N^{2}-N}{3}=\frac{6}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$N^{2}-\frac{1}{3}N=\frac{6}{3}$$

Divide 6 by 3.

$$N^{2}-\frac{1}{3}N=2$$

Divide -\frac{1}{3}, the coefficient of the x term, by 2 to get -\frac{1}{6}. Then add the square of -\frac{1}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$N^{2}-\frac{1}{3}N+\left(-\frac{1}{6}\right)^{2}=2+\left(-\frac{1}{6}\right)^{2}$$

Square -\frac{1}{6} by squaring both the numerator and the denominator of the fraction.

$$N^{2}-\frac{1}{3}N+\frac{1}{36}=2+\frac{1}{36}$$

Add 2 to \frac{1}{36}.

$$N^{2}-\frac{1}{3}N+\frac{1}{36}=\frac{73}{36}$$

Factor N^{2}-\frac{1}{3}N+\frac{1}{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}.

$$\left(N-\frac{1}{6}\right)^{2}=\frac{73}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(N-\frac{1}{6}\right)^{2}}=\sqrt{\frac{73}{36}}$$

Simplify.

$$N-\frac{1}{6}=\frac{\sqrt{73}}{6} N-\frac{1}{6}=-\frac{\sqrt{73}}{6}$$

Add \frac{1}{6} to both sides of the equation.

$$N=\frac{\sqrt{73}+1}{6} N=\frac{1-\sqrt{73}}{6}$$