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

Shown below is the given equation:

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

First, note that on the left side we are able to factor out a common factor from the terms. The largest common factor for the numbers and letters in this case is $x^{99}$ due to the fact that the power of 99 is the lowest power in the equation. Therefore it 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. Hence this is the term with the highest power that can be factored out as a common factor from all letter terms,

Continue to factor the expression:

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

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 given that the only way to obtain a result of 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:

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

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.