Solve 100x^3-180x^2+72x

$$100 x ^ { 3 } - 180 x ^ { 2 } + 72 x$$

Factor

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

Solution Steps

Factor out 4.

$$4\left(25x^{3}-45x^{2}+18x\right)$$

Consider 25x^{3}-45x^{2}+18x. Factor out x.

$$x\left(25x^{2}-45x+18\right)$$

Consider 25x^{2}-45x+18. Factor the expression by grouping. First, the expression needs to be rewritten as 25x^{2}+ax+bx+18. To find a and b, set up a system to be solved.

$$a+b=-45 ab=25\times 18=450$$

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 450.

$$-1,-450 -2,-225 -3,-150 -5,-90 -6,-75 -9,-50 -10,-45 -15,-30 -18,-25$$

Calculate the sum for each pair.

$$-1-450=-451 -2-225=-227 -3-150=-153 -5-90=-95 -6-75=-81 -9-50=-59 -10-45=-55 -15-30=-45 -18-25=-43$$

The solution is the pair that gives sum -45.

$$a=-30 b=-15$$

Rewrite 25x^{2}-45x+18 as \left(25x^{2}-30x\right)+\left(-15x+18\right).

$$\left(25x^{2}-30x\right)+\left(-15x+18\right)$$

Factor out 5x in the first and -3 in the second group.

$$5x\left(5x-6\right)-3\left(5x-6\right)$$

Factor out common term 5x-6 by using distributive property.

$$\left(5x-6\right)\left(5x-3\right)$$

Rewrite the complete factored expression.

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

Evaluate

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