Solve x^2-27/2x+18

$$x ^ { 2 } - \frac { 27 } { 2 } x + 18$$

Factor

$$\frac{\left(x-12\right)\left(2x-3\right)}{2}$$

Solution Steps

Factor out \frac{1}{2}.

$$\frac{2x^{2}-27x+36}{2}$$

Consider 2x^{2}-27x+36. Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx+36. To find a and b, set up a system to be solved.

$$a+b=-27 ab=2\times 36=72$$

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 72.

$$-1,-72 -2,-36 -3,-24 -4,-18 -6,-12 -8,-9$$

Calculate the sum for each pair.

$$-1-72=-73 -2-36=-38 -3-24=-27 -4-18=-22 -6-12=-18 -8-9=-17$$

The solution is the pair that gives sum -27.

$$a=-24 b=-3$$

Rewrite 2x^{2}-27x+36 as \left(2x^{2}-24x\right)+\left(-3x+36\right).

$$\left(2x^{2}-24x\right)+\left(-3x+36\right)$$

Factor out 2x in the first and -3 in the second group.

$$2x\left(x-12\right)-3\left(x-12\right)$$

Factor out common term x-12 by using distributive property.

$$\left(x-12\right)\left(2x-3\right)$$

Rewrite the complete factored expression.

$$\frac{\left(x-12\right)\left(2x-3\right)}{2}$$

Evaluate

$$x^{2}-\frac{27x}{2}+18$$