Use Viette's formulae: " the product of both roots of a quadratic eq. equals its free coefficient divided by its leading coef.,
and the sum of the roots equals minus its linear coef. divided by its leading one".
Thus,
.
Well, now just show that both expressions in the RHS above are equal (Viete again),
and for part (ii) is similar.
Tonio
Thus
If the two roots are and then we know that
Expanding this and equating powers of x from your a, b, c equation form gives
and
Solve the first of these for b and square it:
So
and since
we have, after some simplifying:
You do the l, m, n quadratic the same way.
-Dan
If α and β are the roots of ax^2 + bx + c = 0, then the sum of the roots = α + β = - b/a and product of the roots = αβ = c/a.
In the second equation,
(α^2 + β^2) + (α^2 - β^2) = -p and (α^2 + β^2) * (α^2 - β^2) = q.....(1)
Now (α^2 + β^2) = (α + β)^2 - 2αβ = ...?
(α - β)^2 = (α + β)^2 - 4αβ =.......? Find (α - β)
(α^2 - β^2) = (α + β)(α - β) = ......
Substitute these values in eq.(1)