Solve 3-a-a^2

$$3 - a - a ^ { 2 }$$

Factor

$$-\left(a-\frac{-\sqrt{13}-1}{2}\right)\left(a-\frac{\sqrt{13}-1}{2}\right)$$

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.

$$-a^{2}-a+3=0$$

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.

$$a=\frac{-\left(-1\right)±\sqrt{1-4\left(-1\right)\times 3}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$a=\frac{-\left(-1\right)±\sqrt{1+4\times 3}}{2\left(-1\right)}$$

Multiply 4 times 3.

$$a=\frac{-\left(-1\right)±\sqrt{1+12}}{2\left(-1\right)}$$

Add 1 to 12.

$$a=\frac{-\left(-1\right)±\sqrt{13}}{2\left(-1\right)}$$

The opposite of -1 is 1.

$$a=\frac{1±\sqrt{13}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$a=\frac{1±\sqrt{13}}{-2}$$

Now solve the equation a=\frac{1±\sqrt{13}}{-2} when ± is plus. Add 1 to \sqrt{13}.

$$a=\frac{\sqrt{13}+1}{-2}$$

Divide 1+\sqrt{13} by -2.

$$a=\frac{-\sqrt{13}-1}{2}$$

Now solve the equation a=\frac{1±\sqrt{13}}{-2} when ± is minus. Subtract \sqrt{13} from 1.

$$a=\frac{1-\sqrt{13}}{-2}$$

Divide 1-\sqrt{13} by -2.

$$a=\frac{\sqrt{13}-1}{2}$$

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

$$-a^{2}-a+3=-\left(a-\frac{-\sqrt{13}-1}{2}\right)\left(a-\frac{\sqrt{13}-1}{2}\right)$$

Evaluate

$$3-a-a^{2}$$