Solve x^2+169x-1024

$${ x }^{ 2 } +169x-1024$$

Factor

$$\left(x-\frac{-17\sqrt{113}-169}{2}\right)\left(x-\frac{17\sqrt{113}-169}{2}\right)$$

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.

$$x^{2}+169x-1024=0$$

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.

$$x=\frac{-169±\sqrt{169^{2}-4\left(-1024\right)}}{2}$$

Square 169.

$$x=\frac{-169±\sqrt{28561-4\left(-1024\right)}}{2}$$

Multiply -4 times -1024.

$$x=\frac{-169±\sqrt{28561+4096}}{2}$$

Add 28561 to 4096.

$$x=\frac{-169±\sqrt{32657}}{2}$$

Take the square root of 32657.

$$x=\frac{-169±17\sqrt{113}}{2}$$

Now solve the equation x=\frac{-169±17\sqrt{113}}{2} when ± is plus. Add -169 to 17\sqrt{113}.

$$x=\frac{17\sqrt{113}-169}{2}$$

Now solve the equation x=\frac{-169±17\sqrt{113}}{2} when ± is minus. Subtract 17\sqrt{113} from -169.

$$x=\frac{-17\sqrt{113}-169}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-169+17\sqrt{113}}{2} for x_{1} and \frac{-169-17\sqrt{113}}{2} for x_{2}.

$$x^{2}+169x-1024=\left(x-\frac{17\sqrt{113}-169}{2}\right)\left(x-\frac{-17\sqrt{113}-169}{2}\right)$$

Evaluate

$$x^{2}+169x-1024$$