Solve 3x^2-6x+1

$$3 { x }^{ 2 } -6x+1$$

Factor

$$3\left(x-\left(-\frac{\sqrt{6}}{3}+1\right)\right)\left(x-\left(\frac{\sqrt{6}}{3}+1\right)\right)$$

Steps Using the Quadratic Formula

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.

$$3x^{2}-6x+1=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 3}}{2\times 3}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 3}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-6\right)±\sqrt{36-12}}{2\times 3}$$

Add 36 to -12.

$$x=\frac{-\left(-6\right)±\sqrt{24}}{2\times 3}$$

Take the square root of 24.

$$x=\frac{-\left(-6\right)±2\sqrt{6}}{2\times 3}$$

The opposite of -6 is 6.

$$x=\frac{6±2\sqrt{6}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{6±2\sqrt{6}}{6}$$

Now solve the equation x=\frac{6±2\sqrt{6}}{6} when ± is plus. Add 6 to 2\sqrt{6}.

$$x=\frac{2\sqrt{6}+6}{6}$$

Divide 6+2\sqrt{6} by 6.

$$x=\frac{\sqrt{6}}{3}+1$$

Now solve the equation x=\frac{6±2\sqrt{6}}{6} when ± is minus. Subtract 2\sqrt{6} from 6.

$$x=\frac{6-2\sqrt{6}}{6}$$

Divide 6-2\sqrt{6} by 6.

$$x=-\frac{\sqrt{6}}{3}+1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 1+\frac{\sqrt{6}}{3} for x_{1} and 1-\frac{\sqrt{6}}{3} for x_{2}.

$$3x^{2}-6x+1=3\left(x-\left(\frac{\sqrt{6}}{3}+1\right)\right)\left(x-\left(-\frac{\sqrt{6}}{3}+1\right)\right)$$

Evaluate

$$3x^{2}-6x+1$$