Solve -23x^2+22x+1>0

$$-23 { x }^{ 2 } +22x+1 > 0$$

Solve for x

$$x\in \left(-\frac{1}{23},1\right)$$

Steps Using the Quadratic Formula

Multiply the inequality by -1 to make the coefficient of the highest power in -23x^{2}+22x+1 positive. Since -1 is negative, the inequality direction is changed.

$$23x^{2}-22x-1<0$$

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.

$$23x^{2}-22x-1=0$$

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 23 for a, -22 for b, and -1 for c in the quadratic formula.

$$x=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\times 23\left(-1\right)}}{2\times 23}$$

Do the calculations.

$$x=\frac{22±24}{46}$$

Solve the equation x=\frac{22±24}{46} when ± is plus and when ± is minus.

$$x=1 x=-\frac{1}{23}$$

Rewrite the inequality by using the obtained solutions.

$$23\left(x-1\right)\left(x+\frac{1}{23}\right)<0$$

For the product to be negative, x-1 and x+\frac{1}{23} have to be of the opposite signs. Consider the case when x-1 is positive and x+\frac{1}{23} is negative.

$$x-1>0 x+\frac{1}{23}<0$$

This is false for any x.

$$x\in \emptyset$$

Consider the case when x+\frac{1}{23} is positive and x-1 is negative.

$$x+\frac{1}{23}>0 x-1<0$$

The solution satisfying both inequalities is x\in \left(-\frac{1}{23},1\right).

$$x\in \left(-\frac{1}{23},1\right)$$

The final solution is the union of the obtained solutions.

$$x\in \left(-\frac{1}{23},1\right)$$