If `alpha, beta` be the roots of the equation `ax^2 + bx + c= 0` and `gamma, delta` those of equation `lx^2 + mx + n = 0`,then find the equation whose roots are `alphagamma+betadelta` and `alphadelta+betagamma`

Answer

`a^(2)l^(2)x^(2)-ablmx+(b^(2)-2ac) Ink +(m^(2)-2In)ac=0`

Explanation

We have `alpha + beta=-b/a`
`alpha beta=c/aimpliesgamma + delta =-m/l` and `gamma delta=n/l`
Now sum of the roots
`=(alpha gamma+beta delta)+(alpha delta +beta gamma)=(alpha +beta)gamma +(alpha +beta) delta`
`=(alpha +beta)(gamma+delta)`
`=(-b/a)(-m/l)=(mbl)/(al)`
and product of the roots
`=(alpha gamma+beta delta)(alpha delta+beta gamma)`
`=(alpha^(2)+beta^(2))gamma delta +alpha beta (gamma^(2)+delta^(2))`
`{(alpha +beta)-2alpha beta} gamma beta+alpa beta {(gamma +delta)^(2)-2 gama delta}`
`={(-b/a)^(2)-(2c)/a}b/l+c/a{(m/l)^(2)-(2n)/l}`
`={(b^(2)-2ac)/(a^(2))}b/l+c/a{(m^(2)-2nl)/(l^(2))}=((b^(2)-2ac)ln+(m^(2)-2nl)ac)/(a^(2)l^(2))`
`:.` Required equation is
`x^(2)-((mb)/(al))x+((b^(2)-2ac)ln+(m^(2)-2nl)ac)/(a^(2)+l^(2))=0`
`impliesa^(2)l^(2)x^(2)-mbalc+(b^(2)-2ac)ln+(m^(2)-2nl)ac=0`