Solve 360s^3-780s^2+400s

$$360 s ^ { 3 } - 780 s ^ { 2 } + 400 s$$

Factor

$$20s\left(3s-4\right)\left(6s-5\right)$$

Solution Steps

Factor out 20.

$$20\left(18s^{3}-39s^{2}+20s\right)$$

Consider 18s^{3}-39s^{2}+20s. Factor out s.

$$s\left(18s^{2}-39s+20\right)$$

Consider 18s^{2}-39s+20. Factor the expression by grouping. First, the expression needs to be rewritten as 18s^{2}+as+bs+20. To find a and b, set up a system to be solved.

$$a+b=-39 ab=18\times 20=360$$

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 360.

$$-1,-360 -2,-180 -3,-120 -4,-90 -5,-72 -6,-60 -8,-45 -9,-40 -10,-36 -12,-30 -15,-24 -18,-20$$

Calculate the sum for each pair.

$$-1-360=-361 -2-180=-182 -3-120=-123 -4-90=-94 -5-72=-77 -6-60=-66 -8-45=-53 -9-40=-49 -10-36=-46 -12-30=-42 -15-24=-39 -18-20=-38$$

The solution is the pair that gives sum -39.

$$a=-24 b=-15$$

Rewrite 18s^{2}-39s+20 as \left(18s^{2}-24s\right)+\left(-15s+20\right).

$$\left(18s^{2}-24s\right)+\left(-15s+20\right)$$

Factor out 6s in the first and -5 in the second group.

$$6s\left(3s-4\right)-5\left(3s-4\right)$$

Factor out common term 3s-4 by using distributive property.

$$\left(3s-4\right)\left(6s-5\right)$$

Rewrite the complete factored expression.

$$20s\left(3s-4\right)\left(6s-5\right)$$

Evaluate

$$20s\left(3s-4\right)\left(6s-5\right)$$