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
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3x^{2}-x+4=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.
x=\frac{-\left(-1\right)±\sqrt{1-4\times 3\times 4}}{2\times 3}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -1 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-1\right)±\sqrt{1-12\times 4}}{2\times 3}
Multiply -4 times 3.
x=\frac{-\left(-1\right)±\sqrt{1-48}}{2\times 3}
Multiply -12 times 4.
x=\frac{-\left(-1\right)±\sqrt{-47}}{2\times 3}
Add 1 to -48.
x=\frac{-\left(-1\right)±\sqrt{47}i}{2\times 3}
Take the square root of -47.
x=\frac{1±\sqrt{47}i}{2\times 3}
The opposite of -1 is 1.
x=\frac{1±\sqrt{47}i}{6}
Multiply 2 times 3.
x=\frac{1+\sqrt{47}i}{6}
Now solve the equation x=\frac{1±\sqrt{47}i}{6} when ± is plus. Add 1 to i\sqrt{47}.
x=\frac{-\sqrt{47}i+1}{6}
Now solve the equation x=\frac{1±\sqrt{47}i}{6} when ± is minus. Subtract i\sqrt{47} from 1.
x=\frac{1+\sqrt{47}i}{6} x=\frac{-\sqrt{47}i+1}{6}
The equation is now solved.
3x^{2}-x+4=0
Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.
3x^{2}-x+4-4=-4
Subtract 4 from both sides of the equation.
3x^{2}-x=-4
Subtracting 4 from itself leaves 0.
\frac{3x^{2}-x}{3}=-\frac{4}{3}
Divide both sides by 3.
x^{2}-\frac{1}{3}x=-\frac{4}{3}
Dividing by 3 undoes the multiplication by 3.
x^{2}-\frac{1}{3}x+\left(-\frac{1}{6}\right)^{2}=-\frac{4}{3}+\left(-\frac{1}{6}\right)^{2}
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+\frac{1}{36}=-\frac{4}{3}+\frac{1}{36}
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{47}{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.
\left(x-\frac{1}{6}\right)^{2}=-\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}.
\sqrt{\left(x-\frac{1}{6}\right)^{2}}=\sqrt{-\frac{47}{36}}
Take the square root of both sides of the equation.
x-\frac{1}{6}=\frac{\sqrt{47}i}{6} x-\frac{1}{6}=-\frac{\sqrt{47}i}{6}
Simplify.
x=\frac{1+\sqrt{47}i}{6} x=\frac{-\sqrt{47}i+1}{6}
Add \frac{1}{6} to both sides of the equation.
Examples
Quadratic equation
{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
4 \sin \theta \cos \theta = 2 \sin \theta
Linear equation
y = 3x + 4
Arithmetic
699 * 533
Matrix
\left[ \begin{array} { l l } { 2 } & { 3 } \\ { 5 } & { 4 } \end{array} \right] \left[ \begin{array} { l l l } { 2 } & { 0 } & { 3 } \\ { -1 } & { 1 } & { 5 } \end{array} \right]
Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
\int _ { 0 } ^ { 1 } x e ^ { - x ^ { 2 } } d x
Limits
\lim _{x \rightarrow-3} \frac{x^{2}-9}{x^{2}+2 x-3}