Solve 3x^2-36x+104

$$3 { x }^{ 2 } -36x+104$$

Factor

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

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.

$$3x^{2}-36x+104=0$$

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.

$$x=\frac{-\left(-36\right)±\sqrt{\left(-36\right)^{2}-4\times 3\times 104}}{2\times 3}$$

Square -36.

$$x=\frac{-\left(-36\right)±\sqrt{1296-4\times 3\times 104}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-36\right)±\sqrt{1296-12\times 104}}{2\times 3}$$

Multiply -12 times 104.

$$x=\frac{-\left(-36\right)±\sqrt{1296-1248}}{2\times 3}$$

Add 1296 to -1248.

$$x=\frac{-\left(-36\right)±\sqrt{48}}{2\times 3}$$

Take the square root of 48.

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

The opposite of -36 is 36.

$$x=\frac{36±4\sqrt{3}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{36±4\sqrt{3}}{6}$$

Now solve the equation x=\frac{36±4\sqrt{3}}{6} when ± is plus. Add 36 to 4\sqrt{3}.

$$x=\frac{4\sqrt{3}+36}{6}$$

Divide 36+4\sqrt{3} by 6.

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

Now solve the equation x=\frac{36±4\sqrt{3}}{6} when ± is minus. Subtract 4\sqrt{3} from 36.

$$x=\frac{36-4\sqrt{3}}{6}$$

Divide 36-4\sqrt{3} by 6.

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

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

$$3x^{2}-36x+104=3\left(x-\left(\frac{2\sqrt{3}}{3}+6\right)\right)\left(x-\left(-\frac{2\sqrt{3}}{3}+6\right)\right)$$

Evaluate

$$3x^{2}-36x+104$$