Solve -5x^3+13x^2-6x

$$- 5 x ^ { 3 } + 13 x ^ { 2 } - 6 x$$

Factor

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

Solution Steps

Factor out x.

$$x\left(-5x^{2}+13x-6\right)$$

Consider -5x^{2}+13x-6. Factor the expression by grouping. First, the expression needs to be rewritten as -5x^{2}+ax+bx-6. To find a and b, set up a system to be solved.

$$a+b=13 ab=-5\left(-6\right)=30$$

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 30.

$$1,30 2,15 3,10 5,6$$

Calculate the sum for each pair.

$$1+30=31 2+15=17 3+10=13 5+6=11$$

The solution is the pair that gives sum 13.

$$a=10 b=3$$

Rewrite -5x^{2}+13x-6 as \left(-5x^{2}+10x\right)+\left(3x-6\right).

$$\left(-5x^{2}+10x\right)+\left(3x-6\right)$$

Factor out 5x in the first and -3 in the second group.

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

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

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

Rewrite the complete factored expression.

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

Evaluate

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