Solve -2x^2+36x-23<0

$$- 2 x ^ { 2 } + 36 x - 23 < 0$$

Solve for x

$$x\in \left(-\infty,-\frac{\sqrt{278}}{2}+9\right)\cup \left(\frac{\sqrt{278}}{2}+9,\infty\right)$$

Steps Using the Quadratic Formula

Multiply the inequality by -1 to make the coefficient of the highest power in -2x^{2}+36x-23 positive. Since -1 is negative, the inequality direction is changed.

$$2x^{2}-36x+23>0$$

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.

$$2x^{2}-36x+23=0$$

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, -36 for b, and 23 for c in the quadratic formula.

$$x=\frac{-\left(-36\right)±\sqrt{\left(-36\right)^{2}-4\times 2\times 23}}{2\times 2}$$

Do the calculations.

$$x=\frac{36±2\sqrt{278}}{4}$$

Solve the equation x=\frac{36±2\sqrt{278}}{4} when ± is plus and when ± is minus.

$$x=\frac{\sqrt{278}}{2}+9 x=-\frac{\sqrt{278}}{2}+9$$

Rewrite the inequality by using the obtained solutions.

$$2\left(x-\left(\frac{\sqrt{278}}{2}+9\right)\right)\left(x-\left(-\frac{\sqrt{278}}{2}+9\right)\right)>0$$

For the product to be positive, x-\left(\frac{\sqrt{278}}{2}+9\right) and x-\left(-\frac{\sqrt{278}}{2}+9\right) have to be both negative or both positive. Consider the case when x-\left(\frac{\sqrt{278}}{2}+9\right) and x-\left(-\frac{\sqrt{278}}{2}+9\right) are both negative.

$$x-\left(\frac{\sqrt{278}}{2}+9\right)<0 x-\left(-\frac{\sqrt{278}}{2}+9\right)<0$$

The solution satisfying both inequalities is x<-\frac{\sqrt{278}}{2}+9.

$$x<-\frac{\sqrt{278}}{2}+9$$

Consider the case when x-\left(\frac{\sqrt{278}}{2}+9\right) and x-\left(-\frac{\sqrt{278}}{2}+9\right) are both positive.

$$x-\left(-\frac{\sqrt{278}}{2}+9\right)>0 x-\left(\frac{\sqrt{278}}{2}+9\right)>0$$

The solution satisfying both inequalities is x>\frac{\sqrt{278}}{2}+9.

$$x>\frac{\sqrt{278}}{2}+9$$

The final solution is the union of the obtained solutions.

$$x<-\frac{\sqrt{278}}{2}+9\text{; }x>\frac{\sqrt{278}}{2}+9$$