Question

Form the equation whose roots are the squares of the sum and of the difference of the roots of
$2x^2+2(m+n)x+m^2+n^2=0$

Answer 1

Let the roots of equation $2x^2+2(m+n)x+m^2+n^2=0$ be $a,b$

We have $a+b=-(m+n)$ and $ab=\frac{m^2+n^2}{2}$

Let the roots of equation $x^2+px+q=0$ be $(a+b{)}^2,(a-b{)}^2$

Sum of roots is $-p=(a+b{)}^2+(a-b{)}^2=2(a^2+b^2)=2\left(\right(a+b{)}^2-2ab)=2((m+n{)}^2-m^2-n^2)=4mn$

$\Rightarrow p=-4mn$

product of roots is $q=(a+b{)}^2\times (a-b{)}^2=(m+n{)}^2\times ((m+n{)}^2-2(m^2+n^2\left)\right)=(m+n{)}^2(-(m-n{)}^2)=-(m^2-n^2{)}^2$

Therefore the required equation is $x^2-4mnx-(m^2-n^2{)}^2=0$