Solve the Quadratic Equation: 3x^2+9x=0 Using Common Factor Method

Solve the following problem:

$3x^2+9x=0$

Shown below is the given problem:

$3x^2+9x=0$

First, note that in the left side we are able to factor the expression by using a common factor. The largest common factor for the numbers and letters in this case is $3x$ due to the fact that the first power is the lowest power in the equation and therefore is included both in the term with the second power and in the term with the first power. Any power higher than this is not included in the term with the first power, which is the lowest. Therefore this is the term with the highest power that can be factored out as a common factor from all terms for the letters,

For the numbers, note that 9 is a multiple of 3, therefore it is the largest common factor for the numbers in both terms of the expression,

Let's continue to factor the expression:

$3x^2+9x=0 \\ \downarrow\\ 3x(x+3)=0$

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 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:

$3x=0 \hspace{8pt}\text{/}:3\\ \boxed{x=0}$

In solving the above equation, we divided both sides of the equation by the term with the variable,

Or:

$x+3=0 \\ \boxed{x=-3}$

Let's summarize the solution of the equation:

$3x^2+9x=0 \\ \downarrow\\ 3x(x+3)=0 \\ \downarrow\\ 3x=0 \rightarrow\boxed{ x=0}\\ x+3=0\rightarrow \boxed{x=-3}\\ \downarrow\\ \boxed{x=0,-3}$

Therefore the correct answer is answer C.