Solve -m/2m-1=-3/m

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

Solve for m

$$m=\sqrt{6}+3\approx 5.449489743$$

$$m=3-\sqrt{6}\approx 0.550510257$$

Steps Using the Quadratic Formula

Variable m cannot be equal to any of the values 0,\frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by m\left(2m-1\right), the least common multiple of 2m-1,m.

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

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

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

Use the distributive property to multiply 2m-1 by 3.

$$-m^{2}=-\left(6m-3\right)$$

To find the opposite of 6m-3, find the opposite of each term.

$$-m^{2}=-6m+3$$

Add 6m to both sides.

$$-m^{2}+6m=3$$

Subtract 3 from both sides.

$$-m^{2}+6m-3=0$$

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

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

Square 6.

$$m=\frac{-6±\sqrt{36-4\left(-1\right)\left(-3\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$m=\frac{-6±\sqrt{36+4\left(-3\right)}}{2\left(-1\right)}$$

Multiply 4 times -3.

$$m=\frac{-6±\sqrt{36-12}}{2\left(-1\right)}$$

Add 36 to -12.

$$m=\frac{-6±\sqrt{24}}{2\left(-1\right)}$$

Take the square root of 24.

$$m=\frac{-6±2\sqrt{6}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$m=\frac{-6±2\sqrt{6}}{-2}$$

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

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

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

$$m=3-\sqrt{6}$$

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

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

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

$$m=\sqrt{6}+3$$

The equation is now solved.

$$m=3-\sqrt{6} m=\sqrt{6}+3$$

Steps for Completing the Square

Variable m cannot be equal to any of the values 0,\frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by m\left(2m-1\right), the least common multiple of 2m-1,m.

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

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

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

Use the distributive property to multiply 2m-1 by 3.

$$-m^{2}=-\left(6m-3\right)$$

To find the opposite of 6m-3, find the opposite of each term.

$$-m^{2}=-6m+3$$

Add 6m to both sides.

$$-m^{2}+6m=3$$

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

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

Divide 6 by -1.

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

Divide 3 by -1.

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

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

$$m^{2}-6m+\left(-3\right)^{2}=-3+\left(-3\right)^{2}$$

Square -3.

$$m^{2}-6m+9=-3+9$$

Add -3 to 9.

$$m^{2}-6m+9=6$$

Factor m^{2}-6m+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}.

$$\left(m-3\right)^{2}=6$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-3\right)^{2}}=\sqrt{6}$$

Simplify.

$$m-3=\sqrt{6} m-3=-\sqrt{6}$$

Add 3 to both sides of the equation.

$$m=\sqrt{6}+3 m=3-\sqrt{6}$$