Solve 3x^2-11/2x+3/2

$$3 { x }^{ 2 } - \frac{ 11 }{ 2 } x+ \frac{ 3 }{ 2 }$$

Factor

$$\frac{\left(2x-3\right)\left(3x-1\right)}{2}$$

Solution Steps

Factor out \frac{1}{2}.

$$\frac{6x^{2}-11x+3}{2}$$

Consider 6x^{2}-11x+3. Factor the expression by grouping. First, the expression needs to be rewritten as 6x^{2}+ax+bx+3. To find a and b, set up a system to be solved.

$$a+b=-11 ab=6\times 3=18$$

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 18.

$$-1,-18 -2,-9 -3,-6$$

Calculate the sum for each pair.

$$-1-18=-19 -2-9=-11 -3-6=-9$$

The solution is the pair that gives sum -11.

$$a=-9 b=-2$$

Rewrite 6x^{2}-11x+3 as \left(6x^{2}-9x\right)+\left(-2x+3\right).

$$\left(6x^{2}-9x\right)+\left(-2x+3\right)$$

Factor out 3x in the first and -1 in the second group.

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

Factor out common term 2x-3 by using distributive property.

$$\left(2x-3\right)\left(3x-1\right)$$

Rewrite the complete factored expression.

$$\frac{\left(2x-3\right)\left(3x-1\right)}{2}$$

Evaluate

$$3x^{2}-\frac{11x}{2}+\frac{3}{2}$$