Solve 72x^2y-43xy+6y

$$72 { x }^{ 2 } y-43xy+6y$$

Factor

$$y\left(8x-3\right)\left(9x-2\right)$$

Solution Steps

Factor out y.

$$y\left(72x^{2}-43x+6\right)$$

Consider 72x^{2}-43x+6. Factor the expression by grouping. First, the expression needs to be rewritten as 72x^{2}+ax+bx+6. To find a and b, set up a system to be solved.

$$a+b=-43 ab=72\times 6=432$$

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 432.

$$-1,-432 -2,-216 -3,-144 -4,-108 -6,-72 -8,-54 -9,-48 -12,-36 -16,-27 -18,-24$$

Calculate the sum for each pair.

$$-1-432=-433 -2-216=-218 -3-144=-147 -4-108=-112 -6-72=-78 -8-54=-62 -9-48=-57 -12-36=-48 -16-27=-43 -18-24=-42$$

The solution is the pair that gives sum -43.

$$a=-27 b=-16$$

Rewrite 72x^{2}-43x+6 as \left(72x^{2}-27x\right)+\left(-16x+6\right).

$$\left(72x^{2}-27x\right)+\left(-16x+6\right)$$

Factor out 9x in the first and -2 in the second group.

$$9x\left(8x-3\right)-2\left(8x-3\right)$$

Factor out common term 8x-3 by using distributive property.

$$\left(8x-3\right)\left(9x-2\right)$$

Rewrite the complete factored expression.

$$y\left(8x-3\right)\left(9x-2\right)$$

Evaluate

$$y\left(8x-3\right)\left(9x-2\right)$$