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3x^{2}-24x+40=0
Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.
x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 3\times 40}}{2\times 3}
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(-24\right)±\sqrt{576-4\times 3\times 40}}{2\times 3}
Square -24.
x=\frac{-\left(-24\right)±\sqrt{576-12\times 40}}{2\times 3}
Multiply -4 times 3.
x=\frac{-\left(-24\right)±\sqrt{576-480}}{2\times 3}
Multiply -12 times 40.
x=\frac{-\left(-24\right)±\sqrt{96}}{2\times 3}
Add 576 to -480.
x=\frac{-\left(-24\right)±4\sqrt{6}}{2\times 3}
Take the square root of 96.
x=\frac{24±4\sqrt{6}}{2\times 3}
The opposite of -24 is 24.
x=\frac{24±4\sqrt{6}}{6}
Multiply 2 times 3.
x=\frac{4\sqrt{6}+24}{6}
Now solve the equation x=\frac{24±4\sqrt{6}}{6} when ± is plus. Add 24 to 4\sqrt{6}.
x=\frac{2\sqrt{6}}{3}+4
Divide 24+4\sqrt{6} by 6.
x=\frac{24-4\sqrt{6}}{6}
Now solve the equation x=\frac{24±4\sqrt{6}}{6} when ± is minus. Subtract 4\sqrt{6} from 24.
x=-\frac{2\sqrt{6}}{3}+4
Divide 24-4\sqrt{6} by 6.
3x^{2}-24x+40=3\left(x-\left(\frac{2\sqrt{6}}{3}+4\right)\right)\left(x-\left(-\frac{2\sqrt{6}}{3}+4\right)\right)
Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 4+\frac{2\sqrt{6}}{3} for x_{1} and 4-\frac{2\sqrt{6}}{3} for x_{2}.