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x\left(\frac{3}{4}x-\frac{1}{3}x-\frac{5}{3}\right)=0
Factor out x.
x=0 x=4
To find equation solutions, solve x=0 and \frac{3x}{4}-\frac{x}{3}-\frac{5}{3}=0.
\frac{5}{12}x^{2}-\frac{5}{3}x=0
Combine \frac{3}{4}x^{2} and -\frac{1}{3}x^{2} to get \frac{5}{12}x^{2}.
x=\frac{-\left(-\frac{5}{3}\right)±\sqrt{\left(-\frac{5}{3}\right)^{2}}}{2\times \frac{5}{12}}
This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{5}{12} for a, -\frac{5}{3} for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-\frac{5}{3}\right)±\frac{5}{3}}{2\times \frac{5}{12}}
Take the square root of \left(-\frac{5}{3}\right)^{2}.
x=\frac{\frac{5}{3}±\frac{5}{3}}{2\times \frac{5}{12}}
The opposite of -\frac{5}{3} is \frac{5}{3}.
x=\frac{\frac{5}{3}±\frac{5}{3}}{\frac{5}{6}}
Multiply 2 times \frac{5}{12}.
x=\frac{\frac{10}{3}}{\frac{5}{6}}
Now solve the equation x=\frac{\frac{5}{3}±\frac{5}{3}}{\frac{5}{6}} when ± is plus. Add \frac{5}{3} to \frac{5}{3} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
x=4
Divide \frac{10}{3} by \frac{5}{6} by multiplying \frac{10}{3} by the reciprocal of \frac{5}{6}.
x=\frac{0}{\frac{5}{6}}
Now solve the equation x=\frac{\frac{5}{3}±\frac{5}{3}}{\frac{5}{6}} when ± is minus. Subtract \frac{5}{3} from \frac{5}{3} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.
x=0
Divide 0 by \frac{5}{6} by multiplying 0 by the reciprocal of \frac{5}{6}.
x=4 x=0
The equation is now solved.
\frac{5}{12}x^{2}-\frac{5}{3}x=0
Combine \frac{3}{4}x^{2} and -\frac{1}{3}x^{2} to get \frac{5}{12}x^{2}.
\frac{\frac{5}{12}x^{2}-\frac{5}{3}x}{\frac{5}{12}}=\frac{0}{\frac{5}{12}}
Divide both sides of the equation by \frac{5}{12}, which is the same as multiplying both sides by the reciprocal of the fraction.
x^{2}+\left(-\frac{\frac{5}{3}}{\frac{5}{12}}\right)x=\frac{0}{\frac{5}{12}}
Dividing by \frac{5}{12} undoes the multiplication by \frac{5}{12}.
x^{2}-4x=\frac{0}{\frac{5}{12}}
Divide -\frac{5}{3} by \frac{5}{12} by multiplying -\frac{5}{3} by the reciprocal of \frac{5}{12}.
x^{2}-4x=0
Divide 0 by \frac{5}{12} by multiplying 0 by the reciprocal of \frac{5}{12}.
x^{2}-4x+\left(-2\right)^{2}=\left(-2\right)^{2}
Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-4x+4=4
Square -2.
\left(x-2\right)^{2}=4
Factor x^{2}-4x+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-2\right)^{2}}=\sqrt{4}
Take the square root of both sides of the equation.
x-2=2 x-2=-2
Simplify.
x=4 x=0
Add 2 to both sides of the equation.