Solve F(x)=-x^2+1/2x+1/2

$$F ( x ) = - x ^ { 2 } + \frac { 1 } { 2 } x + \frac { 1 } { 2 }$$

Factor

$$\frac{\left(1-x\right)\left(2x+1\right)}{2}$$

Solution Steps

Factor out \frac{1}{2}.

$$\frac{-2x^{2}+x+1}{2}$$

Consider -2x^{2}+x+1. Factor the expression by grouping. First, the expression needs to be rewritten as -2x^{2}+ax+bx+1. 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 -2x^{2}+x+1 as \left(-2x^{2}+2x\right)+\left(-x+1\right).

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

Factor out 2x in -2x^{2}+2x.

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

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

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

Rewrite the complete factored expression.

$$\frac{\left(-x+1\right)\left(2x+1\right)}{2}$$

Evaluate

$$\frac{1+x-2x^{2}}{2}$$