Solve m^2-left(m-1right)left(3m-2right)>0

$${ m }^{ 2 } - \left( m-1 \right) \left( 3m-2 \right) > 0$$

Solve for m

$$m\in \left(\frac{1}{2},2\right)$$

Steps Using the Quadratic Formula

Use the distributive property to multiply m-1 by 3m-2 and combine like terms.

$$m^{2}-\left(3m^{2}-5m+2\right)>0$$

To find the opposite of 3m^{2}-5m+2, find the opposite of each term.

$$m^{2}-3m^{2}+5m-2>0$$

Combine m^{2} and -3m^{2} to get -2m^{2}.

$$-2m^{2}+5m-2>0$$

Multiply the inequality by -1 to make the coefficient of the highest power in -2m^{2}+5m-2 positive. Since -1 is negative, the inequality direction is changed.

$$2m^{2}-5m+2<0$$

To solve the inequality, factor the left hand side. Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$2m^{2}-5m+2=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 2 for a, -5 for b, and 2 for c in the quadratic formula.

$$m=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 2\times 2}}{2\times 2}$$

Do the calculations.

$$m=\frac{5±3}{4}$$

Solve the equation m=\frac{5±3}{4} when ± is plus and when ± is minus.

$$m=2 m=\frac{1}{2}$$

Rewrite the inequality by using the obtained solutions.

$$2\left(m-2\right)\left(m-\frac{1}{2}\right)<0$$

For the product to be negative, m-2 and m-\frac{1}{2} have to be of the opposite signs. Consider the case when m-2 is positive and m-\frac{1}{2} is negative.

$$m-2>0 m-\frac{1}{2}<0$$

This is false for any m.

$$m\in \emptyset$$

Consider the case when m-\frac{1}{2} is positive and m-2 is negative.

$$m-\frac{1}{2}>0 m-2<0$$

The solution satisfying both inequalities is m\in \left(\frac{1}{2},2\right).

$$m\in \left(\frac{1}{2},2\right)$$

The final solution is the union of the obtained solutions.

$$m\in \left(\frac{1}{2},2\right)$$