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x\left(\frac{1}{12}x-\frac{1}{4}\right)=0
Factor out x.
x=0 x=3
To find equation solutions, solve x=0 and \frac{x}{12}-\frac{1}{4}=0.
\frac{1}{12}x^{2}-\frac{1}{4}x=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(-\frac{1}{4}\right)±\sqrt{\left(-\frac{1}{4}\right)^{2}}}{2\times \frac{1}{12}}
This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{12} for a, -\frac{1}{4} for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-\frac{1}{4}\right)±\frac{1}{4}}{2\times \frac{1}{12}}
Take the square root of \left(-\frac{1}{4}\right)^{2}.
x=\frac{\frac{1}{4}±\frac{1}{4}}{2\times \frac{1}{12}}
The opposite of -\frac{1}{4} is \frac{1}{4}.
x=\frac{\frac{1}{4}±\frac{1}{4}}{\frac{1}{6}}
Multiply 2 times \frac{1}{12}.
x=\frac{\frac{1}{2}}{\frac{1}{6}}
Now solve the equation x=\frac{\frac{1}{4}±\frac{1}{4}}{\frac{1}{6}} when ± is plus. Add \frac{1}{4} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
x=3
Divide \frac{1}{2} by \frac{1}{6} by multiplying \frac{1}{2} by the reciprocal of \frac{1}{6}.
x=\frac{0}{\frac{1}{6}}
Now solve the equation x=\frac{\frac{1}{4}±\frac{1}{4}}{\frac{1}{6}} when ± is minus. Subtract \frac{1}{4} from \frac{1}{4} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.
x=0
Divide 0 by \frac{1}{6} by multiplying 0 by the reciprocal of \frac{1}{6}.
x=3 x=0
The equation is now solved.
\frac{1}{12}x^{2}-\frac{1}{4}x=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}{12}x^{2}-\frac{1}{4}x}{\frac{1}{12}}=\frac{0}{\frac{1}{12}}
Multiply both sides by 12.
x^{2}+\left(-\frac{\frac{1}{4}}{\frac{1}{12}}\right)x=\frac{0}{\frac{1}{12}}
Dividing by \frac{1}{12} undoes the multiplication by \frac{1}{12}.
x^{2}-3x=\frac{0}{\frac{1}{12}}
Divide -\frac{1}{4} by \frac{1}{12} by multiplying -\frac{1}{4} by the reciprocal of \frac{1}{12}.
x^{2}-3x=0
Divide 0 by \frac{1}{12} by multiplying 0 by the reciprocal of \frac{1}{12}.
x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=\left(-\frac{3}{2}\right)^{2}
Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-3x+\frac{9}{4}=\frac{9}{4}
Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.
\left(x-\frac{3}{2}\right)^{2}=\frac{9}{4}
Factor x^{2}-3x+\frac{9}{4}. 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{3}{2}\right)^{2}}=\sqrt{\frac{9}{4}}
Take the square root of both sides of the equation.
x-\frac{3}{2}=\frac{3}{2} x-\frac{3}{2}=-\frac{3}{2}
Simplify.
x=3 x=0
Add \frac{3}{2} to both sides of the equation.