Solve for x
x=\frac{\sqrt{921}}{12}+\frac{1}{4}\approx 2.778998484
x=-\frac{\sqrt{921}}{12}+\frac{1}{4}\approx -2.278998484
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6x^{2}-3x+2=40
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.
6x^{2}-3x+2-40=40-40
Subtract 40 from both sides of the equation.
6x^{2}-3x+2-40=0
Subtracting 40 from itself leaves 0.
6x^{2}-3x-38=0
Subtract 40 from 2.
x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 6\left(-38\right)}}{2\times 6}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -3 for b, and -38 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-3\right)±\sqrt{9-4\times 6\left(-38\right)}}{2\times 6}
Square -3.
x=\frac{-\left(-3\right)±\sqrt{9-24\left(-38\right)}}{2\times 6}
Multiply -4 times 6.
x=\frac{-\left(-3\right)±\sqrt{9+912}}{2\times 6}
Multiply -24 times -38.
x=\frac{-\left(-3\right)±\sqrt{921}}{2\times 6}
Add 9 to 912.
x=\frac{3±\sqrt{921}}{2\times 6}
The opposite of -3 is 3.
x=\frac{3±\sqrt{921}}{12}
Multiply 2 times 6.
x=\frac{\sqrt{921}+3}{12}
Now solve the equation x=\frac{3±\sqrt{921}}{12} when ± is plus. Add 3 to \sqrt{921}.
x=\frac{\sqrt{921}}{12}+\frac{1}{4}
Divide 3+\sqrt{921} by 12.
x=\frac{3-\sqrt{921}}{12}
Now solve the equation x=\frac{3±\sqrt{921}}{12} when ± is minus. Subtract \sqrt{921} from 3.
x=-\frac{\sqrt{921}}{12}+\frac{1}{4}
Divide 3-\sqrt{921} by 12.
x=\frac{\sqrt{921}}{12}+\frac{1}{4} x=-\frac{\sqrt{921}}{12}+\frac{1}{4}
The equation is now solved.
6x^{2}-3x+2=40
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.
6x^{2}-3x+2-2=40-2
Subtract 2 from both sides of the equation.
6x^{2}-3x=40-2
Subtracting 2 from itself leaves 0.
6x^{2}-3x=38
Subtract 2 from 40.
\frac{6x^{2}-3x}{6}=\frac{38}{6}
Divide both sides by 6.
x^{2}+\left(-\frac{3}{6}\right)x=\frac{38}{6}
Dividing by 6 undoes the multiplication by 6.
x^{2}-\frac{1}{2}x=\frac{38}{6}
Reduce the fraction \frac{-3}{6} to lowest terms by extracting and canceling out 3.
x^{2}-\frac{1}{2}x=\frac{19}{3}
Reduce the fraction \frac{38}{6} to lowest terms by extracting and canceling out 2.
x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=\frac{19}{3}+\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}{3}+\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{307}{48}
Add \frac{19}{3} 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{307}{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{307}{48}}
Take the square root of both sides of the equation.
x-\frac{1}{4}=\frac{\sqrt{921}}{12} x-\frac{1}{4}=-\frac{\sqrt{921}}{12}
Simplify.
x=\frac{\sqrt{921}}{12}+\frac{1}{4} x=-\frac{\sqrt{921}}{12}+\frac{1}{4}
Add \frac{1}{4} to both sides of the equation.
Examples
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Matrix
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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
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Limits
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