Solve for x: 7x⁵ - 14x⁴ = 0 Using Common Factor Method

Shown below is the given equation:

$7x^5-14x^4=0$

First, note that on the left side we are able to factor the expression using a common factor. The largest common factor for the numbers and variables in this case is $7x^4$due to the fact that the fourth power is the lowest power in the equation. Therefore it is included both in the term with the fifth power and the term with the fourth power. Any power higher than this is not included in the term with the fourth power, which is the lowest. Hence this is the term with the highest power that can be factored out as a common factor from all terms for the variables,

For the numbers, note that 14 is a multiple of 7, therefore 7 is the largest common factor for the numbers in both terms of the expression,

Let's continue to factor the expression:

$7x^5-14x^4=0 \\ \downarrow\\ 7x^4(x-2)=0$

Proceed to the left side of the equation that we obtained in the last step. There is a multiplication of algebraic expressions and on the right side the number 0. Therefore due to the fact that the only way to obtain 0 from a multiplication is to multiply by 0, at least one of the expressions in the multiplication on the left side must equal zero,

Meaning:

$7x^4=0 \hspace{8pt}\text{/}:7\\ x^4=0 \hspace{8pt}\text{/}\sqrt[4]{\hspace{6pt}}\\ x=\pm0\\ \boxed{x=0}$

In solving the equation above, we first divided both sides of the equation by the term with the variable, and then we took the fourth root of both sides of the equation.

(In this case, taking an even root of the right side of the equation will yield two possibilities - positive and negative however given that we're dealing with zero, we only obtain one answer)

Or:

$x-2=0\\ \boxed{x=2}$

Let's summarize the solution of the equation:

$7x^5-14x^4=0 \\ \downarrow\\ 7x^4(x-2)=0 \\ \downarrow\\ 7x^4=0 \rightarrow\boxed{ x=0}\\ x-2=0\rightarrow \boxed{x=2}\\ \downarrow\\ \boxed{x=0,2}$

Therefore the correct answer is answer A.