Solve -6x^3+48x^2-72x

$$- 6 x ^ { 3 } + 48 x ^ { 2 } - 72 x$$

Factor

$$6x\left(2-x\right)\left(x-6\right)$$

Solution Steps

Factor out 6.

$$6\left(-x^{3}+8x^{2}-12x\right)$$

Consider -x^{3}+8x^{2}-12x. Factor out x.

$$x\left(-x^{2}+8x-12\right)$$

Consider -x^{2}+8x-12. Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-12. To find a and b, set up a system to be solved.

$$a+b=8 ab=-\left(-12\right)=12$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 12.

$$1,12 2,6 3,4$$

Calculate the sum for each pair.

$$1+12=13 2+6=8 3+4=7$$

The solution is the pair that gives sum 8.

$$a=6 b=2$$

Rewrite -x^{2}+8x-12 as \left(-x^{2}+6x\right)+\left(2x-12\right).

$$\left(-x^{2}+6x\right)+\left(2x-12\right)$$

Factor out -x in the first and 2 in the second group.

$$-x\left(x-6\right)+2\left(x-6\right)$$

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

$$\left(x-6\right)\left(-x+2\right)$$

Rewrite the complete factored expression.

$$6x\left(x-6\right)\left(-x+2\right)$$

Evaluate

$$6x\left(2-x\right)\left(x-6\right)$$