Solve 3x^2-36x+106

$$3 { x }^{ 2 } -36x+106$$

Factor

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

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.

$$3x^{2}-36x+106=0$$

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.

$$x=\frac{-\left(-36\right)±\sqrt{\left(-36\right)^{2}-4\times 3\times 106}}{2\times 3}$$

Square -36.

$$x=\frac{-\left(-36\right)±\sqrt{1296-4\times 3\times 106}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-36\right)±\sqrt{1296-12\times 106}}{2\times 3}$$

Multiply -12 times 106.

$$x=\frac{-\left(-36\right)±\sqrt{1296-1272}}{2\times 3}$$

Add 1296 to -1272.

$$x=\frac{-\left(-36\right)±\sqrt{24}}{2\times 3}$$

Take the square root of 24.

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

The opposite of -36 is 36.

$$x=\frac{36±2\sqrt{6}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{36±2\sqrt{6}}{6}$$

Now solve the equation x=\frac{36±2\sqrt{6}}{6} when ± is plus. Add 36 to 2\sqrt{6}.

$$x=\frac{2\sqrt{6}+36}{6}$$

Divide 36+2\sqrt{6} by 6.

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

Now solve the equation x=\frac{36±2\sqrt{6}}{6} when ± is minus. Subtract 2\sqrt{6} from 36.

$$x=\frac{36-2\sqrt{6}}{6}$$

Divide 36-2\sqrt{6} by 6.

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

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

$$3x^{2}-36x+106=3\left(x-\left(\frac{\sqrt{6}}{3}+6\right)\right)\left(x-\left(-\frac{\sqrt{6}}{3}+6\right)\right)$$

Evaluate

$$3x^{2}-36x+106$$