Solve 2-a-a^2

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

Factor

$$\left(1-a\right)\left(a+2\right)$$

Solution Steps

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

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

Factor the expression by grouping. First, the expression needs to be rewritten as -a^{2}+pa+qa+2. To find p and q, set up a system to be solved.

$$p+q=-1 pq=-2=-2$$

Since pq is negative, p and q have the opposite signs. Since p+q is negative, the negative number has greater absolute value than the positive. The only such pair is the system solution.

$$p=1 q=-2$$

Rewrite -a^{2}-a+2 as \left(-a^{2}+a\right)+\left(-2a+2\right).

$$\left(-a^{2}+a\right)+\left(-2a+2\right)$$

Factor out a in the first and 2 in the second group.

$$a\left(-a+1\right)+2\left(-a+1\right)$$

Factor out common term -a+1 by using distributive property.

$$\left(-a+1\right)\left(a+2\right)$$

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.

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

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.

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

Multiply -4 times -1.

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

Multiply 4 times 2.

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

Add 1 to 8.

$$a=\frac{-\left(-1\right)±\sqrt{9}}{2\left(-1\right)}$$

Take the square root of 9.

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

The opposite of -1 is 1.

$$a=\frac{1±3}{2\left(-1\right)}$$

Multiply 2 times -1.

$$a=\frac{1±3}{-2}$$

Now solve the equation a=\frac{1±3}{-2} when ± is plus. Add 1 to 3.

$$a=\frac{4}{-2}$$

Divide 4 by -2.

$$a=-2$$

Now solve the equation a=\frac{1±3}{-2} when ± is minus. Subtract 3 from 1.

$$a=-\frac{2}{-2}$$

Divide -2 by -2.

$$a=1$$

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

$$-a^{2}-a+2=-\left(a-\left(-2\right)\right)\left(a-1\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$-a^{2}-a+2=-\left(a+2\right)\left(a-1\right)$$

Evaluate

$$\left(1-a\right)\left(a+2\right)$$