Chapter 4, Problem 8CR

To determine

The solution of the equation $x^3-6x^2+8x=0$

Answer to Problem 8CR

Solution:

The solutions of the equation $x^3-6x^2+8x=0$ are $x=0,x=2,x=4$

Explanation of Solution

Given information:

The equation $x^3-6x^2+8x=0$

Here, the equation is $x^3-6x^2+8x=0$

Factor out $x$ ,

  $\Rightarrow x\left(x^2-6x+8\right)=0$

  $\Rightarrow x=0$ or $x^2-6x+8=0$

Consider the quadratic equation $x^2-6x+8=0$

Here, the two numbers whose sum is $b=-6$ and whose product is $a\cdot c=8$ are $-4,-2$

So, rewrite the equation as

  $x^2-2x-4x+8=0$

  $\Rightarrow x\left(x-2\right)-4\left(x-2\right)=0$

  $\Rightarrow \left(x-2\right)\left(x-4\right)=0$

  $\Rightarrow x=2$ or $x=4$

Thus, the solutions of the equation $x^3-6x^2+8x=0$ are $x=0,x=2,x=4$