Find the roots of the following function: f(x)= 3*(x-1)^2 - 6.

In this question, we have to find the roots of f(x) or, in other words, its zeros. This means that we have to find all of the values of x for which f(x)=0. Therefore, the first step is to write down our equation, which we will solve: 3*(x-1)^2 - 6 = 0. The second step is to arrange the equation into its simpler form, by moving adding +6 to each side of the equation and then by dividing both sides by 3. Thus, we are left with the following: (x-1)^2 = 2. The third step is to get rid of any powers in the equation. Therefore, we have to take the square root of each side and we are left with two possible solutions: (i) x-1 = sqrt(2) and (ii) x-1 = -sqrt(2). Hence, after rearranging (i) and (ii) we get our two roots: x = sqrt(2) + 1 and x = -sqrt(2) + 1

It is always worth to check that our answer is correct. In this case, let us check whether for x = sqrt(2) + 1, f(x) is indeed equal to 0. Let us plug it into the equation of the function: f(sqrt(2) + 1) = 3*(sqrt(2)+1 - 1)^2 - 6 = 3*(sqrt(2))^2 - 6 = 32 - 6 = 6 - 6 = 0. This confirms that x = sqrt(2) + 1 is a root of f(x). Now let us test x = -sqrt(2) + 1. Let us plug it into the equation of the function: f(-sqrt(2) + 1) = 3(-sqrt(2)+1 - 1)^2 - 6 = 3*(-sqrt(2))^2 - 6 = 3*2 - 6 = 6 - 6 = 0. This confirms that x = -sqrt(2) + 1 is also a root of f(x).