Question

the graph of ( f(x) = x^3 - 6x^2 + 9x ) is shown. based on the graph, what are the solutions of the equation ( x^3 - 6x^2 + 9x = 0 )?
options:
( x = 3 )
( x = -3, 0 )
( x = 0, 3 )
( x = -3, 0, 3 )

Explanation:

Step1: Recall root - graph relation

The solutions of the equation \(f(x) = 0\) are the x - intercepts of the graph of \(y = f(x)\), i.e., the values of \(x\) where the graph intersects the x - axis (\(y = 0\)).

Step2: Identify x - intercepts from the graph

Looking at the given graph of \(f(x)=x^{3}-6x^{2}+9x\), we can see that the graph intersects the x - axis at \(x = 0\) and \(x = 3\) (note that at \(x = 3\), the graph touches the x - axis and turns around, which means \(x = 3\) is a repeated root). Let's also verify by factoring the function:
Factor \(f(x)=x^{3}-6x^{2}+9x\). First, factor out an \(x\): \(f(x)=x(x^{2}-6x + 9)\). Then, factor the quadratic \(x^{2}-6x + 9=(x - 3)^{2}\). So, \(f(x)=x(x - 3)^{2}\). Setting \(f(x)=0\), we have \(x(x - 3)^{2}=0\). Using the zero - product property (if \(ab = 0\), then either \(a = 0\) or \(b = 0\)), we get \(x=0\) or \(x - 3=0\) (twice). So the solutions are \(x = 0\) and \(x = 3\).

Answer:

C. \(x = 0,3\) (assuming the options are labeled as follows: the third option is \(x = 0,3\))