Solve x-2x^2+3x-1>0

$$x - 2 x ^ { 2 } + 3 x - 1 > 0$$

Solve for x

$$x\in \left(-\frac{\sqrt{2}}{2}+1,\frac{\sqrt{2}}{2}+1\right)$$

Steps Using the Quadratic Formula

Combine x and 3x to get 4x.

$$4x-2x^{2}-1>0$$

Multiply the inequality by -1 to make the coefficient of the highest power in 4x-2x^{2}-1 positive. Since -1 is negative, the inequality direction is changed.

$$-4x+2x^{2}+1<0$$

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.

$$-4x+2x^{2}+1=0$$

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, -4 for b, and 1 for c in the quadratic formula.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 2\times 1}}{2\times 2}$$

Do the calculations.

$$x=\frac{4±2\sqrt{2}}{4}$$

Solve the equation x=\frac{4±2\sqrt{2}}{4} when ± is plus and when ± is minus.

$$x=\frac{\sqrt{2}}{2}+1 x=-\frac{\sqrt{2}}{2}+1$$

Rewrite the inequality by using the obtained solutions.

$$2\left(x-\left(\frac{\sqrt{2}}{2}+1\right)\right)\left(x-\left(-\frac{\sqrt{2}}{2}+1\right)\right)<0$$

For the product to be negative, x-\left(\frac{\sqrt{2}}{2}+1\right) and x-\left(-\frac{\sqrt{2}}{2}+1\right) have to be of the opposite signs. Consider the case when x-\left(\frac{\sqrt{2}}{2}+1\right) is positive and x-\left(-\frac{\sqrt{2}}{2}+1\right) is negative.

$$x-\left(\frac{\sqrt{2}}{2}+1\right)>0 x-\left(-\frac{\sqrt{2}}{2}+1\right)<0$$

This is false for any x.

$$x\in \emptyset$$

Consider the case when x-\left(-\frac{\sqrt{2}}{2}+1\right) is positive and x-\left(\frac{\sqrt{2}}{2}+1\right) is negative.

$$x-\left(-\frac{\sqrt{2}}{2}+1\right)>0 x-\left(\frac{\sqrt{2}}{2}+1\right)<0$$

The solution satisfying both inequalities is x\in \left(-\frac{\sqrt{2}}{2}+1,\frac{\sqrt{2}}{2}+1\right).

$$x\in \left(-\frac{\sqrt{2}}{2}+1,\frac{\sqrt{2}}{2}+1\right)$$

The final solution is the union of the obtained solutions.

$$x\in \left(-\frac{\sqrt{2}}{2}+1,\frac{\sqrt{2}}{2}+1\right)$$