Question

How do you write the quadratic equation whose roots are n and -3n?

Answer 1

`x^2+2nx-3n^2 = 0`

Explanation:

Let:

`f(x) = (x-n)(x+3n) = x^2+2nx-3n^2`

Then `f(x) = 0` if and only if at least one of `(x-n) = 0` or `(x+3n) = 0`

That is, if and only if `x=n` or `x=-3n`

So a suitable quadratic equation would be:

`x^2+2nx-3n^2 = 0`

Any quadratic equation in `x` with these zeros is a non-zero scalar multiple of this equation.