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x\left(\frac{1}{3}x-2\right)=0
Factor out x.
x=0 x=6
To find equation solutions, solve x=0 and \frac{x}{3}-2=0.
\frac{1}{3}x^{2}-2x=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(-2\right)±\sqrt{\left(-2\right)^{2}}}{2\times \frac{1}{3}}
This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{3} for a, -2 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-2\right)±2}{2\times \frac{1}{3}}
Take the square root of \left(-2\right)^{2}.
x=\frac{2±2}{2\times \frac{1}{3}}
The opposite of -2 is 2.
x=\frac{2±2}{\frac{2}{3}}
Multiply 2 times \frac{1}{3}.
x=\frac{4}{\frac{2}{3}}
Now solve the equation x=\frac{2±2}{\frac{2}{3}} when ± is plus. Add 2 to 2.
x=6
Divide 4 by \frac{2}{3} by multiplying 4 by the reciprocal of \frac{2}{3}.
x=\frac{0}{\frac{2}{3}}
Now solve the equation x=\frac{2±2}{\frac{2}{3}} when ± is minus. Subtract 2 from 2.
x=0
Divide 0 by \frac{2}{3} by multiplying 0 by the reciprocal of \frac{2}{3}.
x=6 x=0
The equation is now solved.
\frac{1}{3}x^{2}-2x=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.
\frac{\frac{1}{3}x^{2}-2x}{\frac{1}{3}}=\frac{0}{\frac{1}{3}}
Multiply both sides by 3.
x^{2}+\left(-\frac{2}{\frac{1}{3}}\right)x=\frac{0}{\frac{1}{3}}
Dividing by \frac{1}{3} undoes the multiplication by \frac{1}{3}.
x^{2}-6x=\frac{0}{\frac{1}{3}}
Divide -2 by \frac{1}{3} by multiplying -2 by the reciprocal of \frac{1}{3}.
x^{2}-6x=0
Divide 0 by \frac{1}{3} by multiplying 0 by the reciprocal of \frac{1}{3}.
x^{2}-6x+\left(-3\right)^{2}=\left(-3\right)^{2}
Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-6x+9=9
Square -3.
\left(x-3\right)^{2}=9
Factor x^{2}-6x+9. 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-3\right)^{2}}=\sqrt{9}
Take the square root of both sides of the equation.
x-3=3 x-3=-3
Simplify.
x=6 x=0
Add 3 to both sides of the equation.