Solve 12x^3-39x^2+30x

$$12 x ^ { 3 } - 39 x ^ { 2 } + 30 x$$

Factor

$$3x\left(x-2\right)\left(4x-5\right)$$

Solution Steps

Factor out 3.

$$3\left(4x^{3}-13x^{2}+10x\right)$$

Consider 4x^{3}-13x^{2}+10x. Factor out x.

$$x\left(4x^{2}-13x+10\right)$$

Consider 4x^{2}-13x+10. Factor the expression by grouping. First, the expression needs to be rewritten as 4x^{2}+ax+bx+10. To find a and b, set up a system to be solved.

$$a+b=-13 ab=4\times 10=40$$

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 40.

$$-1,-40 -2,-20 -4,-10 -5,-8$$

Calculate the sum for each pair.

$$-1-40=-41 -2-20=-22 -4-10=-14 -5-8=-13$$

The solution is the pair that gives sum -13.

$$a=-8 b=-5$$

Rewrite 4x^{2}-13x+10 as \left(4x^{2}-8x\right)+\left(-5x+10\right).

$$\left(4x^{2}-8x\right)+\left(-5x+10\right)$$

Factor out 4x in the first and -5 in the second group.

$$4x\left(x-2\right)-5\left(x-2\right)$$

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

$$\left(x-2\right)\left(4x-5\right)$$

Rewrite the complete factored expression.

$$3x\left(x-2\right)\left(4x-5\right)$$

Evaluate

$$3x\left(x-2\right)\left(4x-5\right)$$