Solve -9x^2+6x-1leq0

$$- 9 x ^ { 2 } + 6 x - 1 \leq 0$$

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 -9x^{2}+6x-1 positive. Since -1 is negative, the inequality direction is changed.

$$9x^{2}-6x+1\geq 0$$

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.

$$9x^{2}-6x+1=0$$

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 9 for a, -6 for b, and 1 for c in the quadratic formula.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 9\times 1}}{2\times 9}$$

Do the calculations.

$$x=\frac{6±0}{18}$$

Solutions are the same.

$$x=\frac{1}{3}$$

Rewrite the inequality by using the obtained solutions.

$$9\left(x-\frac{1}{3}\right)^{2}\geq 0$$

Inequality holds for x\in \mathrm{R}.

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