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

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

Solve for m

$$m=\frac{3+\sqrt{23}i}{2}\approx 1.5+2.397915762i$$

$$m=\frac{-\sqrt{23}i+3}{2}\approx 1.5-2.397915762i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

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

Add 3 and 1 to get 4.

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

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

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

Subtract 4 from both sides.

$$m+\frac{-m^{2}+m}{2}-4=0$$

Multiply both sides of the equation by 2.

$$2m-m^{2}+m-8=0$$

Reorder the terms.

$$-m^{2}+m+2m-8=0$$

Combine m and 2m to get 3m.

$$-m^{2}+3m-8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 3 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-3±\sqrt{3^{2}-4\left(-1\right)\left(-8\right)}}{2\left(-1\right)}$$

Square 3.

$$m=\frac{-3±\sqrt{9-4\left(-1\right)\left(-8\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$m=\frac{-3±\sqrt{9+4\left(-8\right)}}{2\left(-1\right)}$$

Multiply 4 times -8.

$$m=\frac{-3±\sqrt{9-32}}{2\left(-1\right)}$$

Add 9 to -32.

$$m=\frac{-3±\sqrt{-23}}{2\left(-1\right)}$$

Take the square root of -23.

$$m=\frac{-3±\sqrt{23}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$m=\frac{-3±\sqrt{23}i}{-2}$$

Now solve the equation m=\frac{-3±\sqrt{23}i}{-2} when ± is plus. Add -3 to i\sqrt{23}.

$$m=\frac{-3+\sqrt{23}i}{-2}$$

Divide -3+i\sqrt{23} by -2.

$$m=\frac{-\sqrt{23}i+3}{2}$$

Now solve the equation m=\frac{-3±\sqrt{23}i}{-2} when ± is minus. Subtract i\sqrt{23} from -3.

$$m=\frac{-\sqrt{23}i-3}{-2}$$

Divide -3-i\sqrt{23} by -2.

$$m=\frac{3+\sqrt{23}i}{2}$$

The equation is now solved.

$$m=\frac{-\sqrt{23}i+3}{2} m=\frac{3+\sqrt{23}i}{2}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

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

Add 3 and 1 to get 4.

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

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

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

Multiply both sides of the equation by 2.

$$2m-m^{2}+m=8$$

Reorder the terms.

$$2m+m-m^{2}=8$$

Combine 2m and m to get 3m.

$$3m-m^{2}=8$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-m^{2}+3m=8$$

Divide both sides by -1.

$$\frac{-m^{2}+3m}{-1}=\frac{8}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$m^{2}+\frac{3}{-1}m=\frac{8}{-1}$$

Divide 3 by -1.

$$m^{2}-3m=\frac{8}{-1}$$

Divide 8 by -1.

$$m^{2}-3m=-8$$

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

$$m^{2}-3m+\left(-\frac{3}{2}\right)^{2}=-8+\left(-\frac{3}{2}\right)^{2}$$

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$m^{2}-3m+\frac{9}{4}=-8+\frac{9}{4}$$

Add -8 to \frac{9}{4}.

$$m^{2}-3m+\frac{9}{4}=-\frac{23}{4}$$

Factor m^{2}-3m+\frac{9}{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}.

$$\left(m-\frac{3}{2}\right)^{2}=-\frac{23}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-\frac{3}{2}\right)^{2}}=\sqrt{-\frac{23}{4}}$$

Simplify.

$$m-\frac{3}{2}=\frac{\sqrt{23}i}{2} m-\frac{3}{2}=-\frac{\sqrt{23}i}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$m=\frac{3+\sqrt{23}i}{2} m=\frac{-\sqrt{23}i+3}{2}$$