Solve -6ab^2+75ab-225a

$$- 6 a b ^ { 2 } + 75 a b - 225 a$$

Factor

$$3a\left(5-b\right)\left(2b-15\right)$$

Solution Steps

Factor out 3.

$$3\left(-2ab^{2}+25ab-75a\right)$$

Consider -2ab^{2}+25ab-75a. Factor out a.

$$a\left(-2b^{2}+25b-75\right)$$

Consider -2b^{2}+25b-75. Factor the expression by grouping. First, the expression needs to be rewritten as -2b^{2}+pb+qb-75. To find p and q, set up a system to be solved.

$$p+q=25 pq=-2\left(-75\right)=150$$

Since pq is positive, p and q have the same sign. Since p+q is positive, p and q are both positive. List all such integer pairs that give product 150.

$$1,150 2,75 3,50 5,30 6,25 10,15$$

Calculate the sum for each pair.

$$1+150=151 2+75=77 3+50=53 5+30=35 6+25=31 10+15=25$$

The solution is the pair that gives sum 25.

$$p=15 q=10$$

Rewrite -2b^{2}+25b-75 as \left(-2b^{2}+15b\right)+\left(10b-75\right).

$$\left(-2b^{2}+15b\right)+\left(10b-75\right)$$

Factor out -b in the first and 5 in the second group.

$$-b\left(2b-15\right)+5\left(2b-15\right)$$

Factor out common term 2b-15 by using distributive property.

$$\left(2b-15\right)\left(-b+5\right)$$

Rewrite the complete factored expression.

$$3a\left(2b-15\right)\left(-b+5\right)$$

Evaluate

$$3a\left(5-b\right)\left(2b-15\right)$$