Let alphaα and betaβ be the roots of the equation 2x^2+2(m+n)x+m^2+n^2=02x2+2(m+n)x+m2+n2=0
As such alpha+beta=-2(m+n)/2=-(m+n)α+β=−2m+n2=−(m+n)
and alphaxxbeta=(m^2+n^2)/2α×β=m2+n22
We have to find the equation whose roots are (alpha+beta)^2(α+β)2 and (alpha-beta)^2(α−β)2.
Sum of these roots will be (alpha+beta)^2+(alpha-beta)^2=2(alpha^2+beta^2)(α+β)2+(α−β)2=2(α2+β2)
= 2((alpha+beta)^2-2alphabeta)=2((-(m+n))^2-2(m^2+n^2)/2)2((α+β)2−2αβ)=2((−(m+n))2−2m2+n22)
= 2(m+n)^2-2(m^2+n^2)=4mn2(m+n)2−2(m2+n2)=4mn
Product of these roots will be (alpha+beta)^2xx(alpha-beta)^2(α+β)2×(α−β)2
= (alpha^2-beta^2)^2=(alpha^2+beta^2)^2-4alpha^2beta^2(α2−β2)2=(α2+β2)2−4α2β2
= ((alpha+beta)^2-2alphabeta)^2-4alpha^2beta^2((α+β)2−2αβ)2−4α2β2
= ((-m-n)^2-2(m^2+n^2)/2)^2-4((m^2+n^2)/2)^2((−m−n)2−2m2+n22)2−4(m2+n22)2
= 4m^2n^2-m^4-n^4-2m^2n^2=-m^4-n^4+2m^2n^24m2n2−m4−n4−2m2n2=−m4−n4+2m2n2
= -(m^2-n^2)^2−(m2−n2)2
Hence equation will be
x^2-x2−(sum of roots)x+x+product of roots=0=0 or
x^2-4mnx-(m^2-n^2)^2=0x2−4mnx−(m2−n2)2=0
