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Question

If the equation ax2+bx+c=0 has non-real roots,prove that 1+ca+ba>0

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Solution

The equation ax2+bx+c=0 is non-real
b24ac<0
b2<4ac
|b|<2ac
b<a+c
a+cb>0
a+b+c>0
As A.M> G.M
a+c2>ac
a+c>2ac
a+c2>ac
b<ac
a+b+c>0
aa+ba+ca>0 by dividing by a
1+ba+ca>0
Hence proved


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