Solve 0=m^2-6m-11

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

Solve for m

$$m=2\sqrt{5}+3\approx 7.472135955$$

$$m=3-2\sqrt{5}\approx -1.472135955$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -6 for b, and -11 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square -6.

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

Multiply -4 times -11.

$$m=\frac{-\left(-6\right)±\sqrt{36+44}}{2}$$

Add 36 to 44.

$$m=\frac{-\left(-6\right)±\sqrt{80}}{2}$$

Take the square root of 80.

$$m=\frac{-\left(-6\right)±4\sqrt{5}}{2}$$

The opposite of -6 is 6.

$$m=\frac{6±4\sqrt{5}}{2}$$

Now solve the equation m=\frac{6±4\sqrt{5}}{2} when ± is plus. Add 6 to 4\sqrt{5}.

$$m=\frac{4\sqrt{5}+6}{2}$$

Divide 6+4\sqrt{5} by 2.

$$m=2\sqrt{5}+3$$

Now solve the equation m=\frac{6±4\sqrt{5}}{2} when ± is minus. Subtract 4\sqrt{5} from 6.

$$m=\frac{6-4\sqrt{5}}{2}$$

Divide 6-4\sqrt{5} by 2.

$$m=3-2\sqrt{5}$$

The equation is now solved.

$$m=2\sqrt{5}+3 m=3-2\sqrt{5}$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

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

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

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

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.

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

Square -3.

$$m^{2}-6m+9=11+9$$

Add 11 to 9.

$$m^{2}-6m+9=20$$

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}.

$$\left(m-3\right)^{2}=20$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-3\right)^{2}}=\sqrt{20}$$

Simplify.

$$m-3=2\sqrt{5} m-3=-2\sqrt{5}$$

Add 3 to both sides of the equation.

$$m=2\sqrt{5}+3 m=3-2\sqrt{5}$$