Question No.

Solution:

Given equation has four roots   √e,√f,√g,√h     ( I will refer to alpha, beta ,gamma,delta as e,f,g,h ) 

Now we try to find out the equation that has the roots e,f,g,h.  ( This technique is called transformation.)

x^4+fx^2+gx+h=0

x^4+fx^2+h=(-gx)

Squaring and simplifying we get

 

x^8+2fx^6+(f^2+2h)x^4+(2fh-c^2)x^2+h^2=0

 

Put x^2=y , we get

 

y^4+2fy^3+(f^2+2h)y^2+(2fh-g^2)y+h^2=0

The above equation has the roots e,f,g,h.

 

64efgh-[4sum(ef)-(sum(e))^2]^2

=64h^2-[4(f^2+2h)_(2f)^2]^2

=0

 

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