Solve -x^2/2+3x-5/2

$$\frac{ - { x }^{ 2 } }{ 2 } +3x- \frac{ 5 }{ 2 }$$

Evaluate

$$\frac{\left(1-x\right)\left(x-5\right)}{2}$$

Short Solution Steps

Since \frac{-x^{2}}{2} and \frac{5}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-x^{2}-5}{2}+3x$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3x times \frac{2}{2}.

$$\frac{-x^{2}-5}{2}+\frac{2\times 3x}{2}$$

Since \frac{-x^{2}-5}{2} and \frac{2\times 3x}{2} have the same denominator, add them by adding their numerators.

$$\frac{-x^{2}-5+2\times 3x}{2}$$

Do the multiplications in -x^{2}-5+2\times 3x.

$$\frac{-x^{2}-5+6x}{2}$$

Factor

$$\frac{\left(1-x\right)\left(x-5\right)}{2}$$

Solution Steps

Factor out \frac{1}{2}.

$$\frac{-x^{2}+6x-5}{2}$$

Consider -x^{2}+6x-5. Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-5. To find a and b, set up a system to be solved.

$$a+b=6 ab=-\left(-5\right)=5$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.

$$a=5 b=1$$

Rewrite -x^{2}+6x-5 as \left(-x^{2}+5x\right)+\left(x-5\right).

$$\left(-x^{2}+5x\right)+\left(x-5\right)$$

Factor out -x in -x^{2}+5x.

$$-x\left(x-5\right)+x-5$$

Factor out common term x-5 by using distributive property.

$$\left(x-5\right)\left(-x+1\right)$$

Rewrite the complete factored expression.

$$\frac{\left(x-5\right)\left(-x+1\right)}{2}$$