Solve 11+a-12a-a^2

$$11 + a - 12 a - a ^ { 2 }$$

Factor

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

Steps Using the Quadratic Formula

Combine a and -12a to get -11a.

$$factor(11-11a-a^{2})$$

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.

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

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.

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

Square -11.

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

Multiply -4 times -1.

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

Multiply 4 times 11.

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

Add 121 to 44.

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

The opposite of -11 is 11.

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

Multiply 2 times -1.

$$a=\frac{11±\sqrt{165}}{-2}$$

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

$$a=\frac{\sqrt{165}+11}{-2}$$

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

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

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

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

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

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

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}.

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

Evaluate

$$11-11a-a^{2}$$

Solution Steps

Combine a and -12a to get -11a.

$$11-11a-a^{2}$$