Solve -2m^2+4=-3m

$$- 2 m ^ { 2 } + 4 = - 3 m$$

Solve for m

$$m = \frac{\sqrt{41} + 3}{4} \approx 2.350781059$$

$$m=\frac{3-\sqrt{41}}{4}\approx -0.850781059$$

Steps Using the Quadratic Formula

Add 3m to both sides.

$$-2m^{2}+4+3m=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-2m^{2}+3m+4=0$$

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

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

Square 3.

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

Multiply -4 times -2.

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

Multiply 8 times 4.

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

Add 9 to 32.

$$m=\frac{-3±\sqrt{41}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$m=\frac{-3±\sqrt{41}}{-4}$$

Now solve the equation m=\frac{-3±\sqrt{41}}{-4} when ± is plus. Add -3 to \sqrt{41}.

$$m=\frac{\sqrt{41}-3}{-4}$$

Divide -3+\sqrt{41} by -4.

$$m=\frac{3-\sqrt{41}}{4}$$

Now solve the equation m=\frac{-3±\sqrt{41}}{-4} when ± is minus. Subtract \sqrt{41} from -3.

$$m=\frac{-\sqrt{41}-3}{-4}$$

Divide -3-\sqrt{41} by -4.

$$m=\frac{\sqrt{41}+3}{4}$$

The equation is now solved.

$$m=\frac{3-\sqrt{41}}{4} m=\frac{\sqrt{41}+3}{4}$$

Steps for Completing the Square

Add 3m to both sides.

$$-2m^{2}+4+3m=0$$

Subtract 4 from both sides. Anything subtracted from zero gives its negation.

$$-2m^{2}+3m=-4$$

Divide both sides by -2.

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

Dividing by -2 undoes the multiplication by -2.

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

Divide 3 by -2.

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

Divide -4 by -2.

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

Divide -\frac{3}{2}, the coefficient of the x term, by 2 to get -\frac{3}{4}. Then add the square of -\frac{3}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{3}{4} by squaring both the numerator and the denominator of the fraction.

$$m^{2}-\frac{3}{2}m+\frac{9}{16}=2+\frac{9}{16}$$

Add 2 to \frac{9}{16}.

$$m^{2}-\frac{3}{2}m+\frac{9}{16}=\frac{41}{16}$$

Factor m^{2}-\frac{3}{2}m+\frac{9}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m-\frac{3}{4}\right)^{2}=\frac{41}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-\frac{3}{4}\right)^{2}}=\sqrt{\frac{41}{16}}$$

Simplify.

$$m-\frac{3}{4}=\frac{\sqrt{41}}{4} m-\frac{3}{4}=-\frac{\sqrt{41}}{4}$$

Add \frac{3}{4} to both sides of the equation.

$$m=\frac{\sqrt{41}+3}{4} m=\frac{3-\sqrt{41}}{4}$$