Re: quadratics with related roots



Adam Atkinson wrote:
> I've been helping someone with maths, and some things have cropped
> up which I remember seeing at school but have never used or seen since.
>
> One particular item I'm curious about is questions of the form:
>
> If the roots of the quadratic equation (insert quadratic here) are
> alpha and beta, write down a quadratic whose roots are (two
> formulas involving alpha and beta), without solving for alpha
> and beta.

Perhaps the place to start is this:

Suppose the quadratic is ax^2 + bx + c = a(x^2 + (b/a)x + (c/a))

If alpha and beta are roots, then

(x - alpha)(x - beta) = x^2 + (b/a)x + (c/a)

which leads to:

alpha*beta = c/a

alpha + beta = -b/a


> I've seen cases like asking for roots alpha^2 and beta^2
> or alpha/beta and beta/alpha.

I'm not sure offhand how to solve these, but I'd guess
that suitable manipulation of the above two identities
would lead me to similar ones for

r*s = (expression in a, b, c)
r + s = (expression in a, b, c)

where r = alpha^2, s = beta^2 and from there I can
write down coefficients of my new quadratic.

Actually, just writing that much down I see the solution.

r*s = alpha^2*beta^2 = (alpha*beta)^2 = (c/a)^2

r + s = alpha^2 + beta^2
= alpha^2 + beta^2 + 2*alpha*beta - 2*alpha*beta
= (alpha + beta)^2 - 2*alpha*beta
= (b/a)^2 - 2(c/a)

I know that the quadratic x^2 - (r+s)x + rs has roots
r and s, so I know that the quadratic

x^2 - [(b/a)^2 - 2(c/a)]x + (c/a)^2

has roots alpha^2 and beta^2. Multiplying by a^2, I find
that the quadratic

a^2 x^2 - [b^2 - 2ac] x + c^2

also has roots alpha^2 and beta^2

> I did maths at university and don't recall ever having do to anything
> like this. Is there some field of maths, science or engineering in
> which people need to produce equations with related roots like this?

Number theory?

- Randy

.

Quantcast