Solve for m
m=\frac{\sqrt{10}-1}{3}\approx 0.72075922
m=\frac{-\sqrt{10}-1}{3}\approx -1.387425887
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m=3mm+3\left(m-1\right)
Variable m cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 3m, the least common multiple of 3,m.
m=3m^{2}+3\left(m-1\right)
Multiply m and m to get m^{2}.
m=3m^{2}+3m-3
Use the distributive property to multiply 3 by m-1.
m-3m^{2}=3m-3
Subtract 3m^{2} from both sides.
m-3m^{2}-3m=-3
Subtract 3m from both sides.
-2m-3m^{2}=-3
Combine m and -3m to get -2m.
-2m-3m^{2}+3=0
Add 3 to both sides.
-3m^{2}-2m+3=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(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-3\right)\times 3}}{2\left(-3\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -2 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
m=\frac{-\left(-2\right)±\sqrt{4-4\left(-3\right)\times 3}}{2\left(-3\right)}
Square -2.
m=\frac{-\left(-2\right)±\sqrt{4+12\times 3}}{2\left(-3\right)}
Multiply -4 times -3.
m=\frac{-\left(-2\right)±\sqrt{4+36}}{2\left(-3\right)}
Multiply 12 times 3.
m=\frac{-\left(-2\right)±\sqrt{40}}{2\left(-3\right)}
Add 4 to 36.
m=\frac{-\left(-2\right)±2\sqrt{10}}{2\left(-3\right)}
Take the square root of 40.
m=\frac{2±2\sqrt{10}}{2\left(-3\right)}
The opposite of -2 is 2.
m=\frac{2±2\sqrt{10}}{-6}
Multiply 2 times -3.
m=\frac{2\sqrt{10}+2}{-6}
Now solve the equation m=\frac{2±2\sqrt{10}}{-6} when ± is plus. Add 2 to 2\sqrt{10}.
m=\frac{-\sqrt{10}-1}{3}
Divide 2+2\sqrt{10} by -6.
m=\frac{2-2\sqrt{10}}{-6}
Now solve the equation m=\frac{2±2\sqrt{10}}{-6} when ± is minus. Subtract 2\sqrt{10} from 2.
m=\frac{\sqrt{10}-1}{3}
Divide 2-2\sqrt{10} by -6.
m=\frac{-\sqrt{10}-1}{3} m=\frac{\sqrt{10}-1}{3}
The equation is now solved.
m=3mm+3\left(m-1\right)
Variable m cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 3m, the least common multiple of 3,m.
m=3m^{2}+3\left(m-1\right)
Multiply m and m to get m^{2}.
m=3m^{2}+3m-3
Use the distributive property to multiply 3 by m-1.
m-3m^{2}=3m-3
Subtract 3m^{2} from both sides.
m-3m^{2}-3m=-3
Subtract 3m from both sides.
-2m-3m^{2}=-3
Combine m and -3m to get -2m.
-3m^{2}-2m=-3
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{-3m^{2}-2m}{-3}=-\frac{3}{-3}
Divide both sides by -3.
m^{2}+\left(-\frac{2}{-3}\right)m=-\frac{3}{-3}
Dividing by -3 undoes the multiplication by -3.
m^{2}+\frac{2}{3}m=-\frac{3}{-3}
Divide -2 by -3.
m^{2}+\frac{2}{3}m=1
Divide -3 by -3.
m^{2}+\frac{2}{3}m+\left(\frac{1}{3}\right)^{2}=1+\left(\frac{1}{3}\right)^{2}
Divide \frac{2}{3}, the coefficient of the x term, by 2 to get \frac{1}{3}. Then add the square of \frac{1}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
m^{2}+\frac{2}{3}m+\frac{1}{9}=1+\frac{1}{9}
Square \frac{1}{3} by squaring both the numerator and the denominator of the fraction.
m^{2}+\frac{2}{3}m+\frac{1}{9}=\frac{10}{9}
Add 1 to \frac{1}{9}.
\left(m+\frac{1}{3}\right)^{2}=\frac{10}{9}
Factor m^{2}+\frac{2}{3}m+\frac{1}{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(m+\frac{1}{3}\right)^{2}}=\sqrt{\frac{10}{9}}
Take the square root of both sides of the equation.
m+\frac{1}{3}=\frac{\sqrt{10}}{3} m+\frac{1}{3}=-\frac{\sqrt{10}}{3}
Simplify.
m=\frac{\sqrt{10}-1}{3} m=\frac{-\sqrt{10}-1}{3}
Subtract \frac{1}{3} from both sides of the equation.