Solve -x^2+26x+2

$$- { x }^{ 2 } +26x+2$$

Factor

$$-\left(x-\left(13-3\sqrt{19}\right)\right)\left(x-\left(3\sqrt{19}+13\right)\right)$$

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.

$$-x^{2}+26x+2=0$$

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.

$$x=\frac{-26±\sqrt{26^{2}-4\left(-1\right)\times 2}}{2\left(-1\right)}$$

Square 26.

$$x=\frac{-26±\sqrt{676-4\left(-1\right)\times 2}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-26±\sqrt{676+4\times 2}}{2\left(-1\right)}$$

Multiply 4 times 2.

$$x=\frac{-26±\sqrt{676+8}}{2\left(-1\right)}$$

Add 676 to 8.

$$x=\frac{-26±\sqrt{684}}{2\left(-1\right)}$$

Take the square root of 684.

$$x=\frac{-26±6\sqrt{19}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-26±6\sqrt{19}}{-2}$$

Now solve the equation x=\frac{-26±6\sqrt{19}}{-2} when ± is plus. Add -26 to 6\sqrt{19}.

$$x=\frac{6\sqrt{19}-26}{-2}$$

Divide -26+6\sqrt{19} by -2.

$$x=13-3\sqrt{19}$$

Now solve the equation x=\frac{-26±6\sqrt{19}}{-2} when ± is minus. Subtract 6\sqrt{19} from -26.

$$x=\frac{-6\sqrt{19}-26}{-2}$$

Divide -26-6\sqrt{19} by -2.

$$x=3\sqrt{19}+13$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 13-3\sqrt{19} for x_{1} and 13+3\sqrt{19} for x_{2}.

$$-x^{2}+26x+2=-\left(x-\left(13-3\sqrt{19}\right)\right)\left(x-\left(3\sqrt{19}+13\right)\right)$$

Evaluate

$$2+26x-x^{2}$$