Solve -x^2+11x+11

$$- { x }^{ 2 } +11x+11$$

Factor

$$-\left(x-\frac{11-\sqrt{165}}{2}\right)\left(x-\frac{\sqrt{165}+11}{2}\right)$$

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.

$$-x^{2}+11x+11=0$$

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.

$$x=\frac{-11±\sqrt{11^{2}-4\left(-1\right)\times 11}}{2\left(-1\right)}$$

Square 11.

$$x=\frac{-11±\sqrt{121-4\left(-1\right)\times 11}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-11±\sqrt{121+4\times 11}}{2\left(-1\right)}$$

Multiply 4 times 11.

$$x=\frac{-11±\sqrt{121+44}}{2\left(-1\right)}$$

Add 121 to 44.

$$x=\frac{-11±\sqrt{165}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-11±\sqrt{165}}{-2}$$

Now solve the equation x=\frac{-11±\sqrt{165}}{-2} when ± is plus. Add -11 to \sqrt{165}.

$$x=\frac{\sqrt{165}-11}{-2}$$

Divide -11+\sqrt{165} by -2.

$$x=\frac{11-\sqrt{165}}{2}$$

Now solve the equation x=\frac{-11±\sqrt{165}}{-2} when ± is minus. Subtract \sqrt{165} from -11.

$$x=\frac{-\sqrt{165}-11}{-2}$$

Divide -11-\sqrt{165} by -2.

$$x=\frac{\sqrt{165}+11}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{11-\sqrt{165}}{2} for x_{1} and \frac{11+\sqrt{165}}{2} for x_{2}.

$$-x^{2}+11x+11=-\left(x-\frac{11-\sqrt{165}}{2}\right)\left(x-\frac{\sqrt{165}+11}{2}\right)$$

Evaluate

$$11+11x-x^{2}$$