Solve x^2-3x+1geq0

$$x ^ { 2 } - 3 x + 1 \geq 0$$

Solve for x

$$x\in (-\infty,\frac{3-\sqrt{5}}{2}]\cup [\frac{\sqrt{5}+3}{2},\infty)$$

Steps Using the Quadratic Formula

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.

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

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 1 for a, -3 for b, and 1 for c in the quadratic formula.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 1\times 1}}{2}$$

Do the calculations.

$$x=\frac{3±\sqrt{5}}{2}$$

Solve the equation x=\frac{3±\sqrt{5}}{2} when ± is plus and when ± is minus.

$$x=\frac{\sqrt{5}+3}{2} x=\frac{3-\sqrt{5}}{2}$$

Rewrite the inequality by using the obtained solutions.

$$\left(x-\frac{\sqrt{5}+3}{2}\right)\left(x-\frac{3-\sqrt{5}}{2}\right)\geq 0$$

For the product to be ≥0, x-\frac{\sqrt{5}+3}{2} and x-\frac{3-\sqrt{5}}{2} have to be both ≤0 or both ≥0. Consider the case when x-\frac{\sqrt{5}+3}{2} and x-\frac{3-\sqrt{5}}{2} are both ≤0.

$$x-\frac{\sqrt{5}+3}{2}\leq 0 x-\frac{3-\sqrt{5}}{2}\leq 0$$

The solution satisfying both inequalities is x\leq \frac{3-\sqrt{5}}{2}.

$$x\leq \frac{3-\sqrt{5}}{2}$$

Consider the case when x-\frac{\sqrt{5}+3}{2} and x-\frac{3-\sqrt{5}}{2} are both ≥0.

$$x-\frac{3-\sqrt{5}}{2}\geq 0 x-\frac{\sqrt{5}+3}{2}\geq 0$$

The solution satisfying both inequalities is x\geq \frac{\sqrt{5}+3}{2}.

$$x\geq \frac{\sqrt{5}+3}{2}$$

The final solution is the union of the obtained solutions.

$$x\leq \frac{3-\sqrt{5}}{2}\text{; }x\geq \frac{\sqrt{5}+3}{2}$$