Solve 2m^2+4=-3m

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

Solve for m

$$m=\frac{-3+\sqrt{23}i}{4}\approx -0.75+1.198957881i$$

$$m=\frac{-\sqrt{23}i-3}{4}\approx -0.75-1.198957881i$$

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}.

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

Square 3.

$$m=\frac{-3±\sqrt{9-4\times 2\times 4}}{2\times 2}$$

Multiply -4 times 2.

$$m=\frac{-3±\sqrt{9-8\times 4}}{2\times 2}$$

Multiply -8 times 4.

$$m=\frac{-3±\sqrt{9-32}}{2\times 2}$$

Add 9 to -32.

$$m=\frac{-3±\sqrt{-23}}{2\times 2}$$

Take the square root of -23.

$$m=\frac{-3±\sqrt{23}i}{2\times 2}$$

Multiply 2 times 2.

$$m=\frac{-3±\sqrt{23}i}{4}$$

Now solve the equation m=\frac{-3±\sqrt{23}i}{4} when ± is plus. Add -3 to i\sqrt{23}.

$$m=\frac{-3+\sqrt{23}i}{4}$$

Now solve the equation m=\frac{-3±\sqrt{23}i}{4} when ± is minus. Subtract i\sqrt{23} from -3.

$$m=\frac{-\sqrt{23}i-3}{4}$$

The equation is now solved.

$$m=\frac{-3+\sqrt{23}i}{4} m=\frac{-\sqrt{23}i-3}{4}$$

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 -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}.

$$m^{2}+\frac{3}{2}m+\frac{9}{16}=-\frac{23}{16}$$

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}.

$$\left(m+\frac{3}{4}\right)^{2}=-\frac{23}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m+\frac{3}{4}\right)^{2}}=\sqrt{-\frac{23}{16}}$$

Simplify.

$$m+\frac{3}{4}=\frac{\sqrt{23}i}{4} m+\frac{3}{4}=-\frac{\sqrt{23}i}{4}$$

Subtract \frac{3}{4} from both sides of the equation.

$$m=\frac{-3+\sqrt{23}i}{4} m=\frac{-\sqrt{23}i-3}{4}$$