Solve -2x^2+6x-5<0

$$- 2 x ^ { 2 } + 6 x - 5 < 0$$

Solve for x

$$x\in \mathrm{R}$$

Steps Using the Quadratic Formula

Multiply the inequality by -1 to make the coefficient of the highest power in -2x^{2}+6x-5 positive. Since -1 is negative, the inequality direction is changed.

$$2x^{2}-6x+5>0$$

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.

$$2x^{2}-6x+5=0$$

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, -6 for b, and 5 for c in the quadratic formula.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 2\times 5}}{2\times 2}$$

Do the calculations.

$$x=\frac{6±\sqrt{-4}}{4}$$

Since the square root of a negative number is not defined in the real field, there are no solutions. Expression 2x^{2}-6x+5 has the same sign for any x. To determine the sign, calculate the value of the expression for x=0.

$$2\times 0^{2}-6\times 0+5=5$$

The value of the expression 2x^{2}-6x+5 is always positive. Inequality holds for x\in \mathrm{R}.

$$x\in \mathrm{R}$$