Solve 3x^2-18x+2

$$3 { x }^{ 2 } -18x+2$$

Factor

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

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.

$$3x^{2}-18x+2=0$$

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.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 3\times 2}}{2\times 3}$$

Square -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\times 3\times 2}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-18\right)±\sqrt{324-12\times 2}}{2\times 3}$$

Multiply -12 times 2.

$$x=\frac{-\left(-18\right)±\sqrt{324-24}}{2\times 3}$$

Add 324 to -24.

$$x=\frac{-\left(-18\right)±\sqrt{300}}{2\times 3}$$

Take the square root of 300.

$$x=\frac{-\left(-18\right)±10\sqrt{3}}{2\times 3}$$

The opposite of -18 is 18.

$$x=\frac{18±10\sqrt{3}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{18±10\sqrt{3}}{6}$$

Now solve the equation x=\frac{18±10\sqrt{3}}{6} when ± is plus. Add 18 to 10\sqrt{3}.

$$x=\frac{10\sqrt{3}+18}{6}$$

Divide 18+10\sqrt{3} by 6.

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

Now solve the equation x=\frac{18±10\sqrt{3}}{6} when ± is minus. Subtract 10\sqrt{3} from 18.

$$x=\frac{18-10\sqrt{3}}{6}$$

Divide 18-10\sqrt{3} by 6.

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

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 3+\frac{5\sqrt{3}}{3} for x_{1} and 3-\frac{5\sqrt{3}}{3} for x_{2}.

$$3x^{2}-18x+2=3\left(x-\left(\frac{5\sqrt{3}}{3}+3\right)\right)\left(x-\left(-\frac{5\sqrt{3}}{3}+3\right)\right)$$

Evaluate

$$3x^{2}-18x+2$$