Solve -m^2-m+1=2(-m)

$$- m ^ { 2 } - m + 1 = 2 ( - m )$$

Solve for m

$$m = \frac{\sqrt{5} + 1}{2} \approx 1.618033989$$

$$m=\frac{1-\sqrt{5}}{2}\approx -0.618033989$$

Steps Using the Quadratic Formula

Subtract 2\left(-m\right) from both sides.

$$-m^{2}-m+1-2\left(-m\right)=0$$

Multiply -1 and 2 to get -2.

$$-m^{2}-m+1-2\left(-1\right)m=0$$

Multiply -2 and -1 to get 2.

$$-m^{2}-m+1+2m=0$$

Combine -m and 2m to get m.

$$-m^{2}+m+1=0$$

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

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

Square 1.

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

Multiply -4 times -1.

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

Add 1 to 4.

$$m=\frac{-1±\sqrt{5}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$m=\frac{-1±\sqrt{5}}{-2}$$

Now solve the equation m=\frac{-1±\sqrt{5}}{-2} when ± is plus. Add -1 to \sqrt{5}.

$$m=\frac{\sqrt{5}-1}{-2}$$

Divide -1+\sqrt{5} by -2.

$$m=\frac{1-\sqrt{5}}{2}$$

Now solve the equation m=\frac{-1±\sqrt{5}}{-2} when ± is minus. Subtract \sqrt{5} from -1.

$$m=\frac{-\sqrt{5}-1}{-2}$$

Divide -1-\sqrt{5} by -2.

$$m=\frac{\sqrt{5}+1}{2}$$

The equation is now solved.

$$m=\frac{1-\sqrt{5}}{2} m=\frac{\sqrt{5}+1}{2}$$

Steps for Completing the Square

Subtract 2\left(-m\right) from both sides.

$$-m^{2}-m+1-2\left(-m\right)=0$$

Multiply -1 and 2 to get -2.

$$-m^{2}-m+1-2\left(-1\right)m=0$$

Multiply -2 and -1 to get 2.

$$-m^{2}-m+1+2m=0$$

Combine -m and 2m to get m.

$$-m^{2}+m+1=0$$

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

$$-m^{2}+m=-1$$

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

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

Divide 1 by -1.

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

Divide -1 by -1.

$$m^{2}-m=1$$

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

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

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$m^{2}-m+\frac{1}{4}=1+\frac{1}{4}$$

Add 1 to \frac{1}{4}.

$$m^{2}-m+\frac{1}{4}=\frac{5}{4}$$

Factor m^{2}-m+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m-\frac{1}{2}\right)^{2}=\frac{5}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-\frac{1}{2}\right)^{2}}=\sqrt{\frac{5}{4}}$$

Simplify.

$$m-\frac{1}{2}=\frac{\sqrt{5}}{2} m-\frac{1}{2}=-\frac{\sqrt{5}}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$m=\frac{\sqrt{5}+1}{2} m=\frac{1-\sqrt{5}}{2}$$