Question

Find a quadratic function with zeros at `a+4` and `2a` where `a` is a constant. Write the function in standard form `f(x)=ax^2+bx+c`, and give the values of `a`, `b` and `c`. Assume the leading coefficient is 1?

Answer 1

`f(x) = x^2-(3a+4)x+(2a^2+8a)`

with coefficients `1`, `-(3a+4)` and `2a^2+8a`

Explanation:

A polynomial has a zero `x=alpha` if and only if it has a factor `(x-alpha)`

So given zeros `a+4` and `2a`, the required quadratic must have factors `(x-(a+4))` and `(x-2a)`

So we can write:

`f(x) = (x-(a+4))(x-2a)`

`color(white)(f(x)) = x^2-(3a+4)x+(2a^2+8a)`

So the three coefficients are `1`, `-(3a+4)` and `2a^2+8a`