Solve 2x-1/2x^2-2

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

Factor

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

Solution Steps

Factor out \frac{1}{2}.

$$\frac{4x-x^{2}-4}{2}$$

Consider 4x-x^{2}-4. Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-x^{2}+4x-4$$

Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-4. To find a and b, set up a system to be solved.

$$a+b=4 ab=-\left(-4\right)=4$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 4.

$$1,4 2,2$$

Calculate the sum for each pair.

$$1+4=5 2+2=4$$

The solution is the pair that gives sum 4.

$$a=2 b=2$$

Rewrite -x^{2}+4x-4 as \left(-x^{2}+2x\right)+\left(2x-4\right).

$$\left(-x^{2}+2x\right)+\left(2x-4\right)$$

Factor out -x in the first and 2 in the second group.

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

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

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

Rewrite the complete factored expression.

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

Evaluate

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