10 points! asap algebra ii

let f(x) = x^2-3x. for what values of x is f(f(x)) = f(x)? enter all the solutions, separated by commas.

-1, 0, 3, 4.

Step-by-step explanation:

f(x) = x^2 - 3x

f(f(x)) = (x^2 - 3x)^2 - 3(x^2 - 3x)    (Replacing the x in f(x) by x^2 - 3x).

when f(f(x) = f(x):

(x^2 - 3x)^2 - 3(x^2 - 3x)  = x^2 - 3x

(x^2 - 3x)^2 - 4(x^2 - 3x) = 0

Let y = x^2 - 3x, then

y^2 - 4y = 0

y(y - 4) = 0

y = 0, 4

when x^2 - 3x = 0, x = 0 , 3.

when x^2 - 3x = 4

x^2 - 3x - 4 = 0

(x + 1)(x - 4) = 0

x = -1, 4.