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

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Answer 1 · 2016-06-01T06:49:06

Desired equation is $x^2-4mnx-(m^2-n^2)^2=0$

Let $alpha$ and $beta$ be the roots of the equation $2x^2+2(m+n)x+m^2+n^2=0$

As such $alpha+beta=-2(m+n)/2=-(m+n)$

and $alphaxxbeta=(m^2+n^2)/2$

We have to find the equation whose roots are $(alpha+beta)^2$ and $(alpha-beta)^2$ .

Sum of these roots will be $(alpha+beta)^2+(alpha-beta)^2=2(alpha^2+beta^2)$

= $2((alpha+beta)^2-2alphabeta)=2((-(m+n))^2-2(m^2+n^2)/2)$

= $2(m+n)^2-2(m^2+n^2)=4mn$

Product of these roots will be $(alpha+beta)^2xx(alpha-beta)^2$

= $(alpha^2-beta^2)^2=(alpha^2+beta^2)^2-4alpha^2beta^2$

= $((alpha+beta)^2-2alphabeta)^2-4alpha^2beta^2$

= $((-m-n)^2-2(m^2+n^2)/2)^2-4((m^2+n^2)/2)^2$

= $4m^2n^2-m^4-n^4-2m^2n^2=-m^4-n^4+2m^2n^2$

= $-(m^2-n^2)^2$

Hence equation will be

$x^2-$ (sum of roots) $x+$ product of roots $=0$ or

$x^2-4mnx-(m^2-n^2)^2=0$

Form the quadratic equation whose roots are the squares of the sum of the roots and square of the difference of the root of the equation 2x^2+2(m+n)x+m^2+n^2=02x^2+2(m+n)x+m^2+n^2=0.?