Solve 3m(m-2)-11=0

$$3 m ( m - 2 ) - 11 = 0$$

Solve for m

$$m=\frac{\sqrt{42}}{3}+1\approx 3.160246899$$

$$m=-\frac{\sqrt{42}}{3}+1\approx -1.160246899$$

Steps Using the Quadratic Formula

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

$$3m^{2}-6m-11=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -6 for b, and -11 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 3\left(-11\right)}}{2\times 3}$$

Square -6.

$$m=\frac{-\left(-6\right)±\sqrt{36-4\times 3\left(-11\right)}}{2\times 3}$$

Multiply -4 times 3.

$$m=\frac{-\left(-6\right)±\sqrt{36-12\left(-11\right)}}{2\times 3}$$

Multiply -12 times -11.

$$m=\frac{-\left(-6\right)±\sqrt{36+132}}{2\times 3}$$

Add 36 to 132.

$$m=\frac{-\left(-6\right)±\sqrt{168}}{2\times 3}$$

Take the square root of 168.

$$m=\frac{-\left(-6\right)±2\sqrt{42}}{2\times 3}$$

The opposite of -6 is 6.

$$m=\frac{6±2\sqrt{42}}{2\times 3}$$

Multiply 2 times 3.

$$m=\frac{6±2\sqrt{42}}{6}$$

Now solve the equation m=\frac{6±2\sqrt{42}}{6} when ± is plus. Add 6 to 2\sqrt{42}.

$$m=\frac{2\sqrt{42}+6}{6}$$

Divide 6+2\sqrt{42} by 6.

$$m=\frac{\sqrt{42}}{3}+1$$

Now solve the equation m=\frac{6±2\sqrt{42}}{6} when ± is minus. Subtract 2\sqrt{42} from 6.

$$m=\frac{6-2\sqrt{42}}{6}$$

Divide 6-2\sqrt{42} by 6.

$$m=-\frac{\sqrt{42}}{3}+1$$

The equation is now solved.

$$m=\frac{\sqrt{42}}{3}+1 m=-\frac{\sqrt{42}}{3}+1$$

Steps for Completing the Square

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

$$3m^{2}-6m-11=0$$

Add 11 to both sides. Anything plus zero gives itself.

$$3m^{2}-6m=11$$

Divide both sides by 3.

$$\frac{3m^{2}-6m}{3}=\frac{11}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$m^{2}+\left(-\frac{6}{3}\right)m=\frac{11}{3}$$

Divide -6 by 3.

$$m^{2}-2m=\frac{11}{3}$$

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.

$$m^{2}-2m+1=\frac{11}{3}+1$$

Add \frac{11}{3} to 1.

$$m^{2}-2m+1=\frac{14}{3}$$

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

$$\left(m-1\right)^{2}=\frac{14}{3}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-1\right)^{2}}=\sqrt{\frac{14}{3}}$$

Simplify.

$$m-1=\frac{\sqrt{42}}{3} m-1=-\frac{\sqrt{42}}{3}$$

Add 1 to both sides of the equation.

$$m=\frac{\sqrt{42}}{3}+1 m=-\frac{\sqrt{42}}{3}+1$$