Question

The graph of f(x) = x3 – 6x2 + 9x is shown. Based on the graph, what are the solutions of the equation x3 – 6x2 + 9x = 0? x = 3 x = –3, 0 x = 0, 3 x = –3, 0, 3

Answer 1

hello :
f(x) = x3 – 6x2 + 9x
the solutions of the equation x3 – 6x2 + 9x = 0 are : 0 , 3
because : f(0) 0 and f(3) = 0

Answer 2

The solutions of the equations
`x^3-6x^2+9x=0` is:

x=0 or x=3

Explanation:

The solutions of the graph of the equation are the possible values of x at which the graph meets the x-axis at a point.

The function f(x) is given by:

`f(x)=x^3-6x^2+9x`

The function f(x) could also be written as:

`f(x)=x^3-3x^2-3x^2+9x\\\\i.e.\\\\f(x)=x^2(x-3)-3x(x-3)\\\\i.e.\\\\f(x)=(x^2-3x)(x-3)\\\\i.e.\\\\f(x)=x(x-3)(x-3)`

i.e. If

`f(x)=0`

then

`x(x-3)(x-3)=0\\\\i.e.\\\\x=0\ or\ x=3`

By the help of the graph we may observe that the function meets the x-axis at x=0 or x=3.