Solve =(x^2-24x-9)

$$= ( x ^ { 2 } - 24 x - 9 )$$

Factor

$$\left(x-\left(12-3\sqrt{17}\right)\right)\left(x-\left(3\sqrt{17}+12\right)\right)$$

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.

$$x^{2}-24x-9=0$$

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.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\left(-9\right)}}{2}$$

Square -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\left(-9\right)}}{2}$$

Multiply -4 times -9.

$$x=\frac{-\left(-24\right)±\sqrt{576+36}}{2}$$

Add 576 to 36.

$$x=\frac{-\left(-24\right)±\sqrt{612}}{2}$$

Take the square root of 612.

$$x=\frac{-\left(-24\right)±6\sqrt{17}}{2}$$

The opposite of -24 is 24.

$$x=\frac{24±6\sqrt{17}}{2}$$

Now solve the equation x=\frac{24±6\sqrt{17}}{2} when ± is plus. Add 24 to 6\sqrt{17}.

$$x=\frac{6\sqrt{17}+24}{2}$$

Divide 24+6\sqrt{17} by 2.

$$x=3\sqrt{17}+12$$

Now solve the equation x=\frac{24±6\sqrt{17}}{2} when ± is minus. Subtract 6\sqrt{17} from 24.

$$x=\frac{24-6\sqrt{17}}{2}$$

Divide 24-6\sqrt{17} by 2.

$$x=12-3\sqrt{17}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 12+3\sqrt{17} for x_{1} and 12-3\sqrt{17} for x_{2}.

$$x^{2}-24x-9=\left(x-\left(3\sqrt{17}+12\right)\right)\left(x-\left(12-3\sqrt{17}\right)\right)$$

Evaluate

$$x^{2}-24x-9$$