Solve for x
x=\frac{\sqrt{465}}{12}+\frac{1}{4}\approx 2.046988221
x=-\frac{\sqrt{465}}{12}+\frac{1}{4}\approx -1.546988221
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-6x^{2}+3x+19=0
Swap sides so that all variable terms are on the left hand side.
x=\frac{-3±\sqrt{3^{2}-4\left(-6\right)\times 19}}{2\left(-6\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -6 for a, 3 for b, and 19 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-3±\sqrt{9-4\left(-6\right)\times 19}}{2\left(-6\right)}
Square 3.
x=\frac{-3±\sqrt{9+24\times 19}}{2\left(-6\right)}
Multiply -4 times -6.
x=\frac{-3±\sqrt{9+456}}{2\left(-6\right)}
Multiply 24 times 19.
x=\frac{-3±\sqrt{465}}{2\left(-6\right)}
Add 9 to 456.
x=\frac{-3±\sqrt{465}}{-12}
Multiply 2 times -6.
x=\frac{\sqrt{465}-3}{-12}
Now solve the equation x=\frac{-3±\sqrt{465}}{-12} when ± is plus. Add -3 to \sqrt{465}.
x=-\frac{\sqrt{465}}{12}+\frac{1}{4}
Divide -3+\sqrt{465} by -12.
x=\frac{-\sqrt{465}-3}{-12}
Now solve the equation x=\frac{-3±\sqrt{465}}{-12} when ± is minus. Subtract \sqrt{465} from -3.
x=\frac{\sqrt{465}}{12}+\frac{1}{4}
Divide -3-\sqrt{465} by -12.
x=-\frac{\sqrt{465}}{12}+\frac{1}{4} x=\frac{\sqrt{465}}{12}+\frac{1}{4}
The equation is now solved.
-6x^{2}+3x+19=0
Swap sides so that all variable terms are on the left hand side.
-6x^{2}+3x=-19
Subtract 19 from both sides. Anything subtracted from zero gives its negation.
\frac{-6x^{2}+3x}{-6}=-\frac{19}{-6}
Divide both sides by -6.
x^{2}+\frac{3}{-6}x=-\frac{19}{-6}
Dividing by -6 undoes the multiplication by -6.
x^{2}-\frac{1}{2}x=-\frac{19}{-6}
Reduce the fraction \frac{3}{-6} to lowest terms by extracting and canceling out 3.
x^{2}-\frac{1}{2}x=\frac{19}{6}
Divide -19 by -6.
x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=\frac{19}{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{19}{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{155}{48}
Add \frac{19}{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{155}{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{155}{48}}
Take the square root of both sides of the equation.
x-\frac{1}{4}=\frac{\sqrt{465}}{12} x-\frac{1}{4}=-\frac{\sqrt{465}}{12}
Simplify.
x=\frac{\sqrt{465}}{12}+\frac{1}{4} x=-\frac{\sqrt{465}}{12}+\frac{1}{4}
Add \frac{1}{4} to both sides of the equation.
Examples
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{ 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}