Let f(x)=x^3 - 2x^2 + 5. For which value(s) of x does f(x)=5?

Although it can be tempting to dive straight into such a question, its always useful in mathematics to take a step back and assess the question before scribbling down panicked answers.

For instance, in this case consider f as a piece of machinery (a function) where a number is inputted and another number is outputted. In this example we seek number(s) to input, which correspond to an output of '5'.

So to go about this question, lets initially set f(x)=5, ie x³ - 2x² + 5 = 5. Now since we forced f(x) to equal 5, we simply rearrange to make x the subject, which gives our desired answer(s). Taking away 5 from each side yields x³ - 2x² =0. Now since both terms contain an x², we can factorize this equation, giving x²(x-2)=0. Note that factorizing x² is preferable to dividing through by x² since by dividing we lose information and potential solutions.

So, x²(x-2)=0 implies that x=0 and x=2 are our solutions, as 0²(0-2)=0 and 2²(2-2)=0. Hence the solutions to the problem is x=0 and x=2, only.