Solve 3x^2-12x+11

$$3 { x }^{ 2 } -12x+11$$

Factor

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

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.

$$3x^{2}-12x+11=0$$

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.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 3\times 11}}{2\times 3}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 3\times 11}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-12\right)±\sqrt{144-12\times 11}}{2\times 3}$$

Multiply -12 times 11.

$$x=\frac{-\left(-12\right)±\sqrt{144-132}}{2\times 3}$$

Add 144 to -132.

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

Take the square root of 12.

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

The opposite of -12 is 12.

$$x=\frac{12±2\sqrt{3}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{12±2\sqrt{3}}{6}$$

Now solve the equation x=\frac{12±2\sqrt{3}}{6} when ± is plus. Add 12 to 2\sqrt{3}.

$$x=\frac{2\sqrt{3}+12}{6}$$

Divide 12+2\sqrt{3} by 6.

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

Now solve the equation x=\frac{12±2\sqrt{3}}{6} when ± is minus. Subtract 2\sqrt{3} from 12.

$$x=\frac{12-2\sqrt{3}}{6}$$

Divide 12-2\sqrt{3} by 6.

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

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

$$3x^{2}-12x+11=3\left(x-\left(\frac{\sqrt{3}}{3}+2\right)\right)\left(x-\left(-\frac{\sqrt{3}}{3}+2\right)\right)$$

Evaluate

$$3x^{2}-12x+11$$