Solve x^2-13x+24

$${ x }^{ 2 } -13x+24$$

Factor

$$\left(x-\frac{13-\sqrt{73}}{2}\right)\left(x-\frac{\sqrt{73}+13}{2}\right)$$

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.

$$x^{2}-13x+24=0$$

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.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 24}}{2}$$

Square -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\times 24}}{2}$$

Multiply -4 times 24.

$$x=\frac{-\left(-13\right)±\sqrt{169-96}}{2}$$

Add 169 to -96.

$$x=\frac{-\left(-13\right)±\sqrt{73}}{2}$$

The opposite of -13 is 13.

$$x=\frac{13±\sqrt{73}}{2}$$

Now solve the equation x=\frac{13±\sqrt{73}}{2} when ± is plus. Add 13 to \sqrt{73}.

$$x=\frac{\sqrt{73}+13}{2}$$

Now solve the equation x=\frac{13±\sqrt{73}}{2} when ± is minus. Subtract \sqrt{73} from 13.

$$x=\frac{13-\sqrt{73}}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{13+\sqrt{73}}{2} for x_{1} and \frac{13-\sqrt{73}}{2} for x_{2}.

$$x^{2}-13x+24=\left(x-\frac{\sqrt{73}+13}{2}\right)\left(x-\frac{13-\sqrt{73}}{2}\right)$$

Evaluate

$$x^{2}-13x+24$$