Solve -x^2+3x/6-1/6=0

$$- x ^ { 2 } + \frac { 3 x } { 6 } - \frac { 1 } { 6 } = 0$$

Solve for x (complex solution)

$$x=\frac{\sqrt{15}i}{12}+\frac{1}{4}\approx 0.25+0.322748612i$$

$$x=-\frac{\sqrt{15}i}{12}+\frac{1}{4}\approx 0.25-0.322748612i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 6.

$$-6x^{2}+3x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6 for a, 3 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square 3.

$$x=\frac{-3±\sqrt{9-4\left(-6\right)\left(-1\right)}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-3±\sqrt{9+24\left(-1\right)}}{2\left(-6\right)}$$

Multiply 24 times -1.

$$x=\frac{-3±\sqrt{9-24}}{2\left(-6\right)}$$

Add 9 to -24.

$$x=\frac{-3±\sqrt{-15}}{2\left(-6\right)}$$

Take the square root of -15.

$$x=\frac{-3±\sqrt{15}i}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{-3±\sqrt{15}i}{-12}$$

Now solve the equation x=\frac{-3±\sqrt{15}i}{-12} when ± is plus. Add -3 to i\sqrt{15}.

$$x=\frac{-3+\sqrt{15}i}{-12}$$

Divide -3+i\sqrt{15} by -12.

$$x=-\frac{\sqrt{15}i}{12}+\frac{1}{4}$$

Now solve the equation x=\frac{-3±\sqrt{15}i}{-12} when ± is minus. Subtract i\sqrt{15} from -3.

$$x=\frac{-\sqrt{15}i-3}{-12}$$

Divide -3-i\sqrt{15} by -12.

$$x=\frac{\sqrt{15}i}{12}+\frac{1}{4}$$

The equation is now solved.

$$x=-\frac{\sqrt{15}i}{12}+\frac{1}{4} x=\frac{\sqrt{15}i}{12}+\frac{1}{4}$$

Steps for Completing the Square

Multiply both sides of the equation by 6.

$$-6x^{2}+3x-1=0$$

Add 1 to both sides. Anything plus zero gives itself.

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

Divide both sides by -6.

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

Dividing by -6 undoes the multiplication by -6.

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

Reduce the fraction \frac{3}{-6} to lowest terms by extracting and canceling out 3.

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

Divide 1 by -6.

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

Divide -\frac{1}{2}, the coefficient of the x term, by 2 to get -\frac{1}{4}. Then add the square of -\frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=-\frac{1}{6}+\frac{1}{16}$$

Add -\frac{1}{6} to \frac{1}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=-\frac{5}{48}$$

Factor x^{2}-\frac{1}{2}x+\frac{1}{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}.

$$\left(x-\frac{1}{4}\right)^{2}=-\frac{5}{48}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{-\frac{5}{48}}$$

Simplify.

$$x-\frac{1}{4}=\frac{\sqrt{15}i}{12} x-\frac{1}{4}=-\frac{\sqrt{15}i}{12}$$

Add \frac{1}{4} to both sides of the equation.

$$x=\frac{\sqrt{15}i}{12}+\frac{1}{4} x=-\frac{\sqrt{15}i}{12}+\frac{1}{4}$$