Solve -3=-m^2+m+2

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

Solve for m

$$m = \frac{\sqrt{21} + 1}{2} \approx 2.791287847$$

$$m=\frac{1-\sqrt{21}}{2}\approx -1.791287847$$

Steps Using the Quadratic Formula

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

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

Add 3 to both sides.

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

Add 2 and 3 to get 5.

$$-m^{2}+m+5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 1 for b, and 5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square 1.

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

Multiply -4 times -1.

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

Multiply 4 times 5.

$$m=\frac{-1±\sqrt{1+20}}{2\left(-1\right)}$$

Add 1 to 20.

$$m=\frac{-1±\sqrt{21}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$m=\frac{-1±\sqrt{21}}{-2}$$

Now solve the equation m=\frac{-1±\sqrt{21}}{-2} when ± is plus. Add -1 to \sqrt{21}.

$$m=\frac{\sqrt{21}-1}{-2}$$

Divide -1+\sqrt{21} by -2.

$$m=\frac{1-\sqrt{21}}{2}$$

Now solve the equation m=\frac{-1±\sqrt{21}}{-2} when ± is minus. Subtract \sqrt{21} from -1.

$$m=\frac{-\sqrt{21}-1}{-2}$$

Divide -1-\sqrt{21} by -2.

$$m=\frac{\sqrt{21}+1}{2}$$

The equation is now solved.

$$m=\frac{1-\sqrt{21}}{2} m=\frac{\sqrt{21}+1}{2}$$

Steps for Completing the Square

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

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

Subtract 2 from both sides.

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

Subtract 2 from -3 to get -5.

$$-m^{2}+m=-5$$

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

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

Divide 1 by -1.

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

Divide -5 by -1.

$$m^{2}-m=5$$

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.

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

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

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

Add 5 to \frac{1}{4}.

$$m^{2}-m+\frac{1}{4}=\frac{21}{4}$$

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}.

$$\left(m-\frac{1}{2}\right)^{2}=\frac{21}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-\frac{1}{2}\right)^{2}}=\sqrt{\frac{21}{4}}$$

Simplify.

$$m-\frac{1}{2}=\frac{\sqrt{21}}{2} m-\frac{1}{2}=-\frac{\sqrt{21}}{2}$$

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

$$m=\frac{\sqrt{21}+1}{2} m=\frac{1-\sqrt{21}}{2}$$