Solve x^3-15x^2+50x

$$x ^ { 3 } - 15 x ^ { 2 } + 50 x$$

Factor

$$x\left(x-10\right)\left(x-5\right)$$

Solution Steps

Factor out x.

$$x\left(x^{2}-15x+50\right)$$

Consider x^{2}-15x+50. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+50. To find a and b, set up a system to be solved.

$$a+b=-15 ab=1\times 50=50$$

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 50.

$$-1,-50 -2,-25 -5,-10$$

Calculate the sum for each pair.

$$-1-50=-51 -2-25=-27 -5-10=-15$$

The solution is the pair that gives sum -15.

$$a=-10 b=-5$$

Rewrite x^{2}-15x+50 as \left(x^{2}-10x\right)+\left(-5x+50\right).

$$\left(x^{2}-10x\right)+\left(-5x+50\right)$$

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

$$x\left(x-10\right)-5\left(x-10\right)$$

Factor out common term x-10 by using distributive property.

$$\left(x-10\right)\left(x-5\right)$$

Rewrite the complete factored expression.

$$x\left(x-10\right)\left(x-5\right)$$

Evaluate

$$x\left(x-10\right)\left(x-5\right)$$