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\frac{3}{2}m+m^{2}=2m
Add m^{2} to both sides.
\frac{3}{2}m+m^{2}-2m=0
Subtract 2m from both sides.
-\frac{1}{2}m+m^{2}=0
Combine \frac{3}{2}m and -2m to get -\frac{1}{2}m.
m\left(-\frac{1}{2}+m\right)=0
Factor out m.
m=0 m=\frac{1}{2}
To find equation solutions, solve m=0 and -\frac{1}{2}+m=0.
\frac{3}{2}m+m^{2}=2m
Add m^{2} to both sides.
\frac{3}{2}m+m^{2}-2m=0
Subtract 2m from both sides.
-\frac{1}{2}m+m^{2}=0
Combine \frac{3}{2}m and -2m to get -\frac{1}{2}m.
m^{2}-\frac{1}{2}m=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.
m=\frac{-\left(-\frac{1}{2}\right)±\sqrt{\left(-\frac{1}{2}\right)^{2}}}{2}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -\frac{1}{2} for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
m=\frac{-\left(-\frac{1}{2}\right)±\frac{1}{2}}{2}
Take the square root of \left(-\frac{1}{2}\right)^{2}.
m=\frac{\frac{1}{2}±\frac{1}{2}}{2}
The opposite of -\frac{1}{2} is \frac{1}{2}.
m=\frac{1}{2}
Now solve the equation m=\frac{\frac{1}{2}±\frac{1}{2}}{2} when ± is plus. Add \frac{1}{2} to \frac{1}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
m=\frac{0}{2}
Now solve the equation m=\frac{\frac{1}{2}±\frac{1}{2}}{2} when ± is minus. Subtract \frac{1}{2} from \frac{1}{2} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.
m=0
Divide 0 by 2.
m=\frac{1}{2} m=0
The equation is now solved.
\frac{3}{2}m+m^{2}=2m
Add m^{2} to both sides.
\frac{3}{2}m+m^{2}-2m=0
Subtract 2m from both sides.
-\frac{1}{2}m+m^{2}=0
Combine \frac{3}{2}m and -2m to get -\frac{1}{2}m.
m^{2}-\frac{1}{2}m=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.
m^{2}-\frac{1}{2}m+\left(-\frac{1}{4}\right)^{2}=\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.
m^{2}-\frac{1}{2}m+\frac{1}{16}=\frac{1}{16}
Square -\frac{1}{4} by squaring both the numerator and the denominator of the fraction.
\left(m-\frac{1}{4}\right)^{2}=\frac{1}{16}
Factor m^{2}-\frac{1}{2}m+\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(m-\frac{1}{4}\right)^{2}}=\sqrt{\frac{1}{16}}
Take the square root of both sides of the equation.
m-\frac{1}{4}=\frac{1}{4} m-\frac{1}{4}=-\frac{1}{4}
Simplify.
m=\frac{1}{2} m=0
Add \frac{1}{4} to both sides of the equation.