Solve x^2-24x+12

$${ x }^{ 2 } -24x+12$$

Factor

$$\left(x-\left(12-2\sqrt{33}\right)\right)\left(x-\left(2\sqrt{33}+12\right)\right)$$

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.

$$x^{2}-24x+12=0$$

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.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 12}}{2}$$

Square -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 12}}{2}$$

Multiply -4 times 12.

$$x=\frac{-\left(-24\right)±\sqrt{576-48}}{2}$$

Add 576 to -48.

$$x=\frac{-\left(-24\right)±\sqrt{528}}{2}$$

Take the square root of 528.

$$x=\frac{-\left(-24\right)±4\sqrt{33}}{2}$$

The opposite of -24 is 24.

$$x=\frac{24±4\sqrt{33}}{2}$$

Now solve the equation x=\frac{24±4\sqrt{33}}{2} when ± is plus. Add 24 to 4\sqrt{33}.

$$x=\frac{4\sqrt{33}+24}{2}$$

Divide 24+4\sqrt{33} by 2.

$$x=2\sqrt{33}+12$$

Now solve the equation x=\frac{24±4\sqrt{33}}{2} when ± is minus. Subtract 4\sqrt{33} from 24.

$$x=\frac{24-4\sqrt{33}}{2}$$

Divide 24-4\sqrt{33} by 2.

$$x=12-2\sqrt{33}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 12+2\sqrt{33} for x_{1} and 12-2\sqrt{33} for x_{2}.

$$x^{2}-24x+12=\left(x-\left(2\sqrt{33}+12\right)\right)\left(x-\left(12-2\sqrt{33}\right)\right)$$

Evaluate

$$x^{2}-24x+12$$