Solve 3x^2/2+2x/4+2=0

$$\frac { 3 x ^ { 2 } } { 2 } + \frac { 2 x } { 4 } + 2 = 0$$

Solve for x (complex solution)

$$x=\frac{-1+\sqrt{47}i}{6}\approx -0.166666667+1.1426091i$$

$$x=\frac{-\sqrt{47}i-1}{6}\approx -0.166666667-1.1426091i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 4, the least common multiple of 2,4.

$$2\times 3x^{2}+2x+8=0$$

Multiply 2 and 3 to get 6.

$$6x^{2}+2x+8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 2 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\times 6\times 8}}{2\times 6}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\times 6\times 8}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-2±\sqrt{4-24\times 8}}{2\times 6}$$

Multiply -24 times 8.

$$x=\frac{-2±\sqrt{4-192}}{2\times 6}$$

Add 4 to -192.

$$x=\frac{-2±\sqrt{-188}}{2\times 6}$$

Take the square root of -188.

$$x=\frac{-2±2\sqrt{47}i}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-2±2\sqrt{47}i}{12}$$

Now solve the equation x=\frac{-2±2\sqrt{47}i}{12} when ± is plus. Add -2 to 2i\sqrt{47}.

$$x=\frac{-2+2\sqrt{47}i}{12}$$

Divide -2+2i\sqrt{47} by 12.

$$x=\frac{-1+\sqrt{47}i}{6}$$

Now solve the equation x=\frac{-2±2\sqrt{47}i}{12} when ± is minus. Subtract 2i\sqrt{47} from -2.

$$x=\frac{-2\sqrt{47}i-2}{12}$$

Divide -2-2i\sqrt{47} by 12.

$$x=\frac{-\sqrt{47}i-1}{6}$$

The equation is now solved.

$$x=\frac{-1+\sqrt{47}i}{6} x=\frac{-\sqrt{47}i-1}{6}$$

Steps for Completing the Square

Multiply both sides of the equation by 4, the least common multiple of 2,4.

$$2\times 3x^{2}+2x+8=0$$

Multiply 2 and 3 to get 6.

$$6x^{2}+2x+8=0$$

Subtract 8 from both sides. Anything subtracted from zero gives its negation.

$$6x^{2}+2x=-8$$

Divide both sides by 6.

$$\frac{6x^{2}+2x}{6}=-\frac{8}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\frac{2}{6}x=-\frac{8}{6}$$

Reduce the fraction \frac{2}{6} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{1}{3}x=-\frac{8}{6}$$

Reduce the fraction \frac{-8}{6} to lowest terms by extracting and canceling out 2.

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

Divide \frac{1}{3}, the coefficient of the x term, by 2 to get \frac{1}{6}. Then add the square of \frac{1}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{3}x+\left(\frac{1}{6}\right)^{2}=-\frac{4}{3}+\left(\frac{1}{6}\right)^{2}$$

Square \frac{1}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{3}x+\frac{1}{36}=-\frac{4}{3}+\frac{1}{36}$$

Add -\frac{4}{3} to \frac{1}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{3}x+\frac{1}{36}=-\frac{47}{36}$$

Factor x^{2}+\frac{1}{3}x+\frac{1}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{6}\right)^{2}=-\frac{47}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{6}\right)^{2}}=\sqrt{-\frac{47}{36}}$$

Simplify.

$$x+\frac{1}{6}=\frac{\sqrt{47}i}{6} x+\frac{1}{6}=-\frac{\sqrt{47}i}{6}$$

Subtract \frac{1}{6} from both sides of the equation.

$$x=\frac{-1+\sqrt{47}i}{6} x=\frac{-\sqrt{47}i-1}{6}$$