Solve Quadratic Function with Factor Theorem

[Rowah]

This is the problem:

A quadratic function f(x) with integral coefficients has the following properties: $$f(3/2)=0$$, (x-2) is a factor of f(x), and f(4) = 50. Determine f(x).

The answer in the back of the book is $$f(x)=5(2x-3)(x-2)$$

I can easily understand the (2x-3) and (x-2), but I don't understand the "5", and "f(4) = 50".

 

[cepheid]

f(4) = 50 means that the quadratic function evaluated at x = 4 has a value of 50. Your final solution must satisfy this condition as well as the others.

Hint, the "5" probably has something to do with that last criterion.

 

[Rowah]

Hmm, when you sub f(4) = 50 in f(x)=(2x-3)(x-2)

You end up with 50=10


Am I on the right track towards implementing that "5" into my final equation?

 

[HallsofIvy]

Which tells you that f(x) is NOT (2x-3)(x-2)!

But you also know that 2x-3 and x- 2 are the only factors involving x.
What happens if you substitute x= 4 into f(x)= A(2x-3)(x-2) where A is a constant?

 

[Rowah]

One word to describe HallsofIvy.. Brilliant!

You end up with A=5, thanks I understand it now :D