Solve 4x^2-25/6x+1

$$4 { x }^{ 2 } - \frac{ 25 }{ 6 } x+1$$

Factor

$$\frac{\left(3x-2\right)\left(8x-3\right)}{6}$$

Solution Steps

Factor out \frac{1}{6}.

$$\frac{24x^{2}-25x+6}{6}$$

Consider 24x^{2}-25x+6. Factor the expression by grouping. First, the expression needs to be rewritten as 24x^{2}+ax+bx+6. To find a and b, set up a system to be solved.

$$a+b=-25 ab=24\times 6=144$$

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 144.

$$-1,-144 -2,-72 -3,-48 -4,-36 -6,-24 -8,-18 -9,-16 -12,-12$$

Calculate the sum for each pair.

$$-1-144=-145 -2-72=-74 -3-48=-51 -4-36=-40 -6-24=-30 -8-18=-26 -9-16=-25 -12-12=-24$$

The solution is the pair that gives sum -25.

$$a=-16 b=-9$$

Rewrite 24x^{2}-25x+6 as \left(24x^{2}-16x\right)+\left(-9x+6\right).

$$\left(24x^{2}-16x\right)+\left(-9x+6\right)$$

Factor out 8x in the first and -3 in the second group.

$$8x\left(3x-2\right)-3\left(3x-2\right)$$

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

$$\left(3x-2\right)\left(8x-3\right)$$

Rewrite the complete factored expression.

$$\frac{\left(3x-2\right)\left(8x-3\right)}{6}$$

Evaluate

$$4x^{2}-\frac{25x}{6}+1$$