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Solve for x (complex solution)
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6x^{2}-3x+1=0
Swap sides so that all variable terms are on the left hand side.
x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 6}}{2\times 6}
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{-\left(-3\right)±\sqrt{9-4\times 6}}{2\times 6}
Square -3.
x=\frac{-\left(-3\right)±\sqrt{9-24}}{2\times 6}
Multiply -4 times 6.
x=\frac{-\left(-3\right)±\sqrt{-15}}{2\times 6}
Add 9 to -24.
x=\frac{-\left(-3\right)±\sqrt{15}i}{2\times 6}
Take the square root of -15.
x=\frac{3±\sqrt{15}i}{2\times 6}
The opposite of -3 is 3.
x=\frac{3±\sqrt{15}i}{12}
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{\sqrt{15}i}{12}+\frac{1}{4}
Divide 3+i\sqrt{15} by 12.
x=\frac{-\sqrt{15}i+3}{12}
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}{12}+\frac{1}{4}
Divide 3-i\sqrt{15} by 12.
x=\frac{\sqrt{15}i}{12}+\frac{1}{4} x=-\frac{\sqrt{15}i}{12}+\frac{1}{4}
The equation is now solved.
6x^{2}-3x+1=0
Swap sides so that all variable terms are on the left hand side.
6x^{2}-3x=-1
Subtract 1 from both sides. Anything subtracted from zero gives its negation.
\frac{6x^{2}-3x}{6}=-\frac{1}{6}
Divide both sides by 6.
x^{2}+\left(-\frac{3}{6}\right)x=-\frac{1}{6}
Dividing by 6 undoes the multiplication by 6.
x^{2}-\frac{1}{2}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+\left(-\frac{1}{4}\right)^{2}=-\frac{1}{6}+\left(-\frac{1}{4}\right)^{2}
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+\frac{1}{16}=-\frac{1}{6}+\frac{1}{16}
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{5}{48}
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.
\left(x-\frac{1}{4}\right)^{2}=-\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}.
\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{-\frac{5}{48}}
Take the square root of both sides of the equation.
x-\frac{1}{4}=\frac{\sqrt{15}i}{12} x-\frac{1}{4}=-\frac{\sqrt{15}i}{12}
Simplify.
x=\frac{\sqrt{15}i}{12}+\frac{1}{4} x=-\frac{\sqrt{15}i}{12}+\frac{1}{4}
Add \frac{1}{4} to both sides of the equation.