Solve 1/3=m+m-1/m

$$\frac{1}{3}=m+\frac{m-1}{m}$$

Solve for m

$$m=\frac{\sqrt{10}-1}{3}\approx 0.72075922$$

$$m=\frac{-\sqrt{10}-1}{3}\approx -1.387425887$$

Steps Using the Quadratic Formula

Variable m cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 3m, the least common multiple of 3,m.

$$m=3mm+3\left(m-1\right)$$

Multiply m and m to get m^{2}.

$$m=3m^{2}+3\left(m-1\right)$$

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

$$m=3m^{2}+3m-3$$

Subtract 3m^{2} from both sides.

$$m-3m^{2}=3m-3$$

Subtract 3m from both sides.

$$m-3m^{2}-3m=-3$$

Combine m and -3m to get -2m.

$$-2m-3m^{2}=-3$$

Add 3 to both sides.

$$-2m-3m^{2}+3=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-3m^{2}-2m+3=0$$

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

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

Square -2.

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

Multiply -4 times -3.

$$m=\frac{-\left(-2\right)±\sqrt{4+12\times 3}}{2\left(-3\right)}$$

Multiply 12 times 3.

$$m=\frac{-\left(-2\right)±\sqrt{4+36}}{2\left(-3\right)}$$

Add 4 to 36.

$$m=\frac{-\left(-2\right)±\sqrt{40}}{2\left(-3\right)}$$

Take the square root of 40.

$$m=\frac{-\left(-2\right)±2\sqrt{10}}{2\left(-3\right)}$$

The opposite of -2 is 2.

$$m=\frac{2±2\sqrt{10}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$m=\frac{2±2\sqrt{10}}{-6}$$

Now solve the equation m=\frac{2±2\sqrt{10}}{-6} when ± is plus. Add 2 to 2\sqrt{10}.

$$m=\frac{2\sqrt{10}+2}{-6}$$

Divide 2+2\sqrt{10} by -6.

$$m=\frac{-\sqrt{10}-1}{3}$$

Now solve the equation m=\frac{2±2\sqrt{10}}{-6} when ± is minus. Subtract 2\sqrt{10} from 2.

$$m=\frac{2-2\sqrt{10}}{-6}$$

Divide 2-2\sqrt{10} by -6.

$$m=\frac{\sqrt{10}-1}{3}$$

The equation is now solved.

$$m=\frac{-\sqrt{10}-1}{3} m=\frac{\sqrt{10}-1}{3}$$

Steps for Completing the Square

Variable m cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 3m, the least common multiple of 3,m.

$$m=3mm+3\left(m-1\right)$$

Multiply m and m to get m^{2}.

$$m=3m^{2}+3\left(m-1\right)$$

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

$$m=3m^{2}+3m-3$$

Subtract 3m^{2} from both sides.

$$m-3m^{2}=3m-3$$

Subtract 3m from both sides.

$$m-3m^{2}-3m=-3$$

Combine m and -3m to get -2m.

$$-2m-3m^{2}=-3$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-3m^{2}-2m=-3$$

Divide both sides by -3.

$$\frac{-3m^{2}-2m}{-3}=-\frac{3}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$m^{2}+\left(-\frac{2}{-3}\right)m=-\frac{3}{-3}$$

Divide -2 by -3.

$$m^{2}+\frac{2}{3}m=-\frac{3}{-3}$$

Divide -3 by -3.

$$m^{2}+\frac{2}{3}m=1$$

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

$$m^{2}+\frac{2}{3}m+\left(\frac{1}{3}\right)^{2}=1+\left(\frac{1}{3}\right)^{2}$$

Square \frac{1}{3} by squaring both the numerator and the denominator of the fraction.

$$m^{2}+\frac{2}{3}m+\frac{1}{9}=1+\frac{1}{9}$$

Add 1 to \frac{1}{9}.

$$m^{2}+\frac{2}{3}m+\frac{1}{9}=\frac{10}{9}$$

Factor m^{2}+\frac{2}{3}m+\frac{1}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m+\frac{1}{3}\right)^{2}=\frac{10}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m+\frac{1}{3}\right)^{2}}=\sqrt{\frac{10}{9}}$$

Simplify.

$$m+\frac{1}{3}=\frac{\sqrt{10}}{3} m+\frac{1}{3}=-\frac{\sqrt{10}}{3}$$

Subtract \frac{1}{3} from both sides of the equation.

$$m=\frac{\sqrt{10}-1}{3} m=\frac{-\sqrt{10}-1}{3}$$