Solve A+2-A^2

$$A + 2 - A ^ { 2 }$$

Factor

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

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}+aA+bA+2. To find a and b, set up a system to be solved.

$$a+b=1 ab=-2=-2$$

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

$$a=2 b=-1$$

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

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

Factor out -A in the first and -1 in the second group.

$$-A\left(A-2\right)-\left(A-2\right)$$

Factor out common term A-2 by using distributive property.

$$\left(A-2\right)\left(-A-1\right)$$

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.

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

Square 1.

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

Multiply -4 times -1.

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

Multiply 4 times 2.

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

Add 1 to 8.

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

Take the square root of 9.

$$A=\frac{-1±3}{2\left(-1\right)}$$

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.

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

Divide 2 by -2.

$$A=-1$$

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

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

Divide -4 by -2.

$$A=2$$

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

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

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

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

Evaluate

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