Chapter 7.8, Problem 23E

To determine

The value n from the given equation using factoring equation is to determine.

Answer to Problem 23E

By factorization Method or using Quadratic formula the value of variable is $n=0,2,4$

Explanation of Solution

Given information: - The given factoring equation are

  $5n^3-30n^2+40n=0$

Calculation: - According to the question

  $5n^3-30n^2+40n=0$

  $5n(n^2-6n+8)=0$ --------------------------------- (1)

Then, factories middle term

  $\begin{array}{l}{n}^2-6n+8=0\\ n^2-4n-2n+8=0\\ n(n-4)-2(n-4)=0\\ n-4=0,n-2=0\\ n=4,n=2\end{array}$

Thus, the value of variable n is 2,4.

And, $5n=0$ [from equation 1]

  $n=0$

By Quadratic equation- According to the Quadratic Equation formula,

  $A{n}^2+Bn+c=0$

Where A, B & C are numbers called coefficient

  $n=\frac{-B\pm \sqrt{B^2-4AC}}{2A}$

In given case,

A = 1

B = $-6$

C = 8

Accordingly

  $\begin{array}{l}{B}^2-4AC\\ $ {(-6)}^2-4\times 1\times 8$36-32=0\\ 4=0\\ B^2-4AC=4\end{array}$

By applying the Quadratic formula

  $n=\frac{-(-6)\pm \sqrt{4}}{2\times 1}$

  $=\frac{6\pm 2}{2}$

  $\frac{6+2}{2}=\frac{8}{2}=4$

  $\frac{6-2}{2}=\frac{4}{2}=2$

and

  $5n=0$

  $n=0$

Thus, the value of variable n is =0,2,4.