Solve 3x^2-12x+4

$$3 { x }^{ 2 } -12x+4$$

Factor

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

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.

$$3x^{2}-12x+4=0$$

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.

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

Square -12.

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

Multiply -4 times 3.

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

Multiply -12 times 4.

$$x=\frac{-\left(-12\right)±\sqrt{144-48}}{2\times 3}$$

Add 144 to -48.

$$x=\frac{-\left(-12\right)±\sqrt{96}}{2\times 3}$$

Take the square root of 96.

$$x=\frac{-\left(-12\right)±4\sqrt{6}}{2\times 3}$$

The opposite of -12 is 12.

$$x=\frac{12±4\sqrt{6}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{12±4\sqrt{6}}{6}$$

Now solve the equation x=\frac{12±4\sqrt{6}}{6} when ± is plus. Add 12 to 4\sqrt{6}.

$$x=\frac{4\sqrt{6}+12}{6}$$

Divide 12+4\sqrt{6} by 6.

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

Now solve the equation x=\frac{12±4\sqrt{6}}{6} when ± is minus. Subtract 4\sqrt{6} from 12.

$$x=\frac{12-4\sqrt{6}}{6}$$

Divide 12-4\sqrt{6} by 6.

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

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

$$3x^{2}-12x+4=3\left(x-\left(\frac{2\sqrt{6}}{3}+2\right)\right)\left(x-\left(-\frac{2\sqrt{6}}{3}+2\right)\right)$$

Evaluate

$$3x^{2}-12x+4$$