Solve 2ax^3-28ax^2+90ax

$$2 a x ^ { 3 } - 28 a x ^ { 2 } + 90 a x$$

Factor

$$2ax\left(x-9\right)\left(x-5\right)$$

Solution Steps

Factor out 2.

$$2\left(ax^{3}-14ax^{2}+45ax\right)$$

Consider ax^{3}-14ax^{2}+45ax. Factor out ax.

$$ax\left(x^{2}-14x+45\right)$$

Consider x^{2}-14x+45. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+px+qx+45. To find p and q, set up a system to be solved.

$$p+q=-14 pq=1\times 45=45$$

Since pq is positive, p and q have the same sign. Since p+q is negative, p and q are both negative. List all such integer pairs that give product 45.

$$-1,-45 -3,-15 -5,-9$$

Calculate the sum for each pair.

$$-1-45=-46 -3-15=-18 -5-9=-14$$

The solution is the pair that gives sum -14.

$$p=-9 q=-5$$

Rewrite x^{2}-14x+45 as \left(x^{2}-9x\right)+\left(-5x+45\right).

$$\left(x^{2}-9x\right)+\left(-5x+45\right)$$

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

$$x\left(x-9\right)-5\left(x-9\right)$$

Factor out common term x-9 by using distributive property.

$$\left(x-9\right)\left(x-5\right)$$

Rewrite the complete factored expression.

$$2ax\left(x-9\right)\left(x-5\right)$$

Evaluate

$$2ax\left(x-9\right)\left(x-5\right)$$