Solve 4x^3-130x^2+900x

$$4 x ^ { 3 } - 130 x ^ { 2 } + 900 x$$

Factor

$$2x\left(x-10\right)\left(2x-45\right)$$

Solution Steps

Factor out 2.

$$2\left(2x^{3}-65x^{2}+450x\right)$$

Consider 2x^{3}-65x^{2}+450x. Factor out x.

$$x\left(2x^{2}-65x+450\right)$$

Consider 2x^{2}-65x+450. Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx+450. To find a and b, set up a system to be solved.

$$a+b=-65 ab=2\times 450=900$$

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 900.

$$-1,-900 -2,-450 -3,-300 -4,-225 -5,-180 -6,-150 -9,-100 -10,-90 -12,-75 -15,-60 -18,-50 -20,-45 -25,-36 -30,-30$$

Calculate the sum for each pair.

$$-1-900=-901 -2-450=-452 -3-300=-303 -4-225=-229 -5-180=-185 -6-150=-156 -9-100=-109 -10-90=-100 -12-75=-87 -15-60=-75 -18-50=-68 -20-45=-65 -25-36=-61 -30-30=-60$$

The solution is the pair that gives sum -65.

$$a=-45 b=-20$$

Rewrite 2x^{2}-65x+450 as \left(2x^{2}-45x\right)+\left(-20x+450\right).

$$\left(2x^{2}-45x\right)+\left(-20x+450\right)$$

Factor out x in the first and -10 in the second group.

$$x\left(2x-45\right)-10\left(2x-45\right)$$

Factor out common term 2x-45 by using distributive property.

$$\left(2x-45\right)\left(x-10\right)$$

Rewrite the complete factored expression.

$$2x\left(2x-45\right)\left(x-10\right)$$

Evaluate

$$2x\left(x-10\right)\left(2x-45\right)$$