If α, β be the roots of ax^2 + bx + c = 0 and γ, δ those of lx^2 + mx + n = 0, find the equation whose roots are (αγ + βδ) and (αδ + βγ)

Roots of the equation ax^2 + bx + c = 0 are α, β

Roots of the equation lx^2 + mx + n = 0 are γ, δ

By applying Sum and Product of roots formulae on both the equations, we will get,

α + β = -b/a          (1)

αβ = c/a                (2)

γ + δ = -m/l          (3)

γδ = n/l                 (4)

We have to find the equation whose roots are (αγ + βδ)  and (αδ + βγ). It has two roots, so obviously it is a quadratic equation. As we know, any quadratic equation can be written as x^2 - (sum of its roots) + product of its roots = 0. So, to find the equation which has roots (αγ + βδ)  and (αδ + βγ), we'll first have to find sum and product of those roots.

Sum of roots

= (αγ + βδ) + (αδ + βγ)

= αγ + αδ + βδ + βγ

= α(γ + δ) + β(γ + δ)

= (α + β)(γ + δ)

= (-b/a)(-m/l)               ( By using (1) and (3) )

= bm/al

Product of roots

= (αγ + βδ) * (αδ + βγ)

= α^2.γδ + αβ.γ^2 + αβ.δ^2 + β^2.γδ

= γδ(α^2 + β^2) + αβ(γ^2 + δ^2)

= γδ[(α + β)^2 - 2αβ] + αβ[(γ + δ)^2 - 2γδ]

Now, by using all (1), (2), (3) and (4)

= (n/l) [(-b/a)^2 - 2(c/a)] + (c/a) [(-m/l)^2 - 2(n/l)]

= (n/l) (b^2/a^2 - 2c/a) + (c/a) (m^2/l^2 - 2n/l)

= (n/l) (b^2 - 2ac)/a^2 + (c/a) (m^2 - 2nl)/l^2

= (n.b^2 - 2acn)/l.a^2 + (c.m^2 - 2cnl)/a.l^2

= (nlb^2 - 2acnl + acm^2 - 2acnl) / a^2.l^2

= (nlb^2 + acm^2 - 4acnl) / a^2.l^2

This is the simplest form in which it can be written. This was more of a simplification job.

So, we've found out that the sum of roots of the desired equation is (bm/al) and the product of roots of the same is (nlb^2 + acm^2 - 4acnl) / a^2.l^2 

Thus, the desired equation should be

x^2 - (bm/al)x + [(nlb^2 + acm^2 - 4acnl) / a^2.l^2] = 0

=> (a^2.l^2)x^2 - (a^2.l^2)(bm/al)x + (nlb^2 + acm^2 - 4acnl) = 0

=> (a^2.l^2)x^2 - (ablm)x + (nlb^2 + acm^2 - 4acnl) = 0