Solve x^2-48x+144

$${ x }^{ 2 } -48x+144$$

Factor

$$\left(x-\left(24-12\sqrt{3}\right)\right)\left(x-\left(12\sqrt{3}+24\right)\right)$$

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.

$$x^{2}-48x+144=0$$

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.

$$x=\frac{-\left(-48\right)±\sqrt{\left(-48\right)^{2}-4\times 144}}{2}$$

Square -48.

$$x=\frac{-\left(-48\right)±\sqrt{2304-4\times 144}}{2}$$

Multiply -4 times 144.

$$x=\frac{-\left(-48\right)±\sqrt{2304-576}}{2}$$

Add 2304 to -576.

$$x=\frac{-\left(-48\right)±\sqrt{1728}}{2}$$

Take the square root of 1728.

$$x=\frac{-\left(-48\right)±24\sqrt{3}}{2}$$

The opposite of -48 is 48.

$$x=\frac{48±24\sqrt{3}}{2}$$

Now solve the equation x=\frac{48±24\sqrt{3}}{2} when ± is plus. Add 48 to 24\sqrt{3}.

$$x=\frac{24\sqrt{3}+48}{2}$$

Divide 48+24\sqrt{3} by 2.

$$x=12\sqrt{3}+24$$

Now solve the equation x=\frac{48±24\sqrt{3}}{2} when ± is minus. Subtract 24\sqrt{3} from 48.

$$x=\frac{48-24\sqrt{3}}{2}$$

Divide 48-24\sqrt{3} by 2.

$$x=24-12\sqrt{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 24+12\sqrt{3} for x_{1} and 24-12\sqrt{3} for x_{2}.

$$x^{2}-48x+144=\left(x-\left(12\sqrt{3}+24\right)\right)\left(x-\left(24-12\sqrt{3}\right)\right)$$

Evaluate

$$x^{2}-48x+144$$