Solve the Equation: x^100 - 9x^99 = 0 Using Common Factoring

The equation in the problem is:

$x^{100}-9x^{99}=0$

First, note that in the left side we can factor out a common factor from the terms, the largest common factor for the numbers and letters in this case is $x^{99}$because the power of 99 is the lowest power in the equation and therefore is included both in the term with power of 100 and in the term with power of 99. Any power higher than this is not included in the term with the lowest power of 99, and therefore this is the term with the highest power that can be factored out as a common factor from all letter terms,

Let's continue and perform the factoring:

$x^{100}-9x^{99}=0 \\ \downarrow\\ x^{99}(x-9)=0$

Let's continue and address the fact that on the left side of the equation we received in the last step there is a multiplication of algebraic expressions and on the right side the number 0, therefore, since the only way to get a result of 0 from multiplication is to multiply by 0, at least one of the expressions in the multiplication on the left side must equal zero,

Meaning:

$x^{99}=0 \hspace{8pt}\text{/}\sqrt[99]{\hspace{6pt}}\\ \boxed{x=0}$

In solving the equation above, we extracted a 99th root from both sides of the equation.

(In this case, extracting an odd root from the right side of the equation yielded one possibility)

Or:

$x-9=0 \\ \boxed{x=9}$

Let's summarize the solution of the equation:

$x^{100}-9x^{99}=0 \\ \downarrow\\ x^{99}(x-9)=0 \\ \downarrow\\ x^{99}=0 \rightarrow\boxed{ x=0}\\ x-9=0\rightarrow \boxed{x=9}\\ \downarrow\\ \boxed{x=0,9}$

Therefore, the correct answer is answer C.