Solve 0=1/2x^2+2x+3

$$0 = \frac { 1 } { 2 } x ^ { 2 } + 2 x + 3$$

Solve for x (complex solution)

$$x=-2+\sqrt{2}i\approx -2+1.414213562i$$

$$x=-\sqrt{2}i-2\approx -2-1.414213562i$$

Steps Using the Quadratic Formula

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

$$\frac{1}{2}x^{2}+2x+3=0$$

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

$$x=\frac{-2±\sqrt{2^{2}-4\times \frac{1}{2}\times 3}}{2\times \frac{1}{2}}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\times \frac{1}{2}\times 3}}{2\times \frac{1}{2}}$$

Multiply -4 times \frac{1}{2}.

$$x=\frac{-2±\sqrt{4-2\times 3}}{2\times \frac{1}{2}}$$

Multiply -2 times 3.

$$x=\frac{-2±\sqrt{4-6}}{2\times \frac{1}{2}}$$

Add 4 to -6.

$$x=\frac{-2±\sqrt{-2}}{2\times \frac{1}{2}}$$

Take the square root of -2.

$$x=\frac{-2±\sqrt{2}i}{2\times \frac{1}{2}}$$

Multiply 2 times \frac{1}{2}.

$$x=\frac{-2±\sqrt{2}i}{1}$$

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

$$x=\frac{-2+\sqrt{2}i}{1}$$

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

$$x=-2+\sqrt{2}i$$

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

$$x=\frac{-\sqrt{2}i-2}{1}$$

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

$$x=-\sqrt{2}i-2$$

The equation is now solved.

$$x=-2+\sqrt{2}i x=-\sqrt{2}i-2$$

Steps for Completing the Square

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

$$\frac{1}{2}x^{2}+2x+3=0$$

Subtract 3 from both sides. Anything subtracted from zero gives its negation.

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

Multiply both sides by 2.

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

Dividing by \frac{1}{2} undoes the multiplication by \frac{1}{2}.

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

Divide 2 by \frac{1}{2} by multiplying 2 by the reciprocal of \frac{1}{2}.

$$x^{2}+4x=-\frac{3}{\frac{1}{2}}$$

Divide -3 by \frac{1}{2} by multiplying -3 by the reciprocal of \frac{1}{2}.

$$x^{2}+4x=-6$$

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

$$x^{2}+4x+2^{2}=-6+2^{2}$$

Square 2.

$$x^{2}+4x+4=-6+4$$

Add -6 to 4.

$$x^{2}+4x+4=-2$$

Factor x^{2}+4x+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}.

$$\left(x+2\right)^{2}=-2$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{-2}$$

Simplify.

$$x+2=\sqrt{2}i x+2=-\sqrt{2}i$$

Subtract 2 from both sides of the equation.

$$x=-2+\sqrt{2}i x=-\sqrt{2}i-2$$