Solve f(x)=x^2/32-x+255/32

$$f ( x ) = \frac { x ^ { 2 } } { 32 } - x + \frac { 255 } { 32 }$$

Evaluate

$$\frac{\left(x-17\right)\left(x-15\right)}{32}$$

Short Solution Steps

Since \frac{x^{2}}{32} and \frac{255}{32} have the same denominator, add them by adding their numerators.

$$\frac{x^{2}+255}{32}-x$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{32}{32}.

$$\frac{x^{2}+255}{32}-\frac{32x}{32}$$

Since \frac{x^{2}+255}{32} and \frac{32x}{32} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x^{2}+255-32x}{32}$$

Factor

$$\frac{\left(x-17\right)\left(x-15\right)}{32}$$

Solution Steps

Factor out \frac{1}{32}.

$$\frac{x^{2}-32x+255}{32}$$

Consider x^{2}-32x+255. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+255. To find a and b, set up a system to be solved.

$$a+b=-32 ab=1\times 255=255$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 255.

$$-1,-255 -3,-85 -5,-51 -15,-17$$

Calculate the sum for each pair.

$$-1-255=-256 -3-85=-88 -5-51=-56 -15-17=-32$$

The solution is the pair that gives sum -32.

$$a=-17 b=-15$$

Rewrite x^{2}-32x+255 as \left(x^{2}-17x\right)+\left(-15x+255\right).

$$\left(x^{2}-17x\right)+\left(-15x+255\right)$$

Factor out x in the first and -15 in the second group.

$$x\left(x-17\right)-15\left(x-17\right)$$

Factor out common term x-17 by using distributive property.

$$\left(x-17\right)\left(x-15\right)$$

Rewrite the complete factored expression.

$$\frac{\left(x-17\right)\left(x-15\right)}{32}$$