We have #a,b,c,dinRR# such that #ab=2(c+d)#.How to prove that at least one of the equations #x^2+ax+c=0; x^2+bx+d=0# have double roots?

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Answer 1 · 2018-01-17T19:33:20

The assertion is false.

Consider the two quadratic equations:

#x^2+ax+c = x^2-5x+6 = (x-2)(x-3) = 0#

and

#x^2+bx+d = x^2-2x-1 = (x-1-sqrt(2))(x-1+sqrt(2)) = 0#

Then:

#ab = (-5)(-2) = 10 = 2(6-1) = 2(c+d)#

Both equations have distinct real roots and:

#ab = 2(c+d)#

So the assertion is false.