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Question

If the equation ax2+2bx+c=0 has real roots, a,b,c being real numbers and if m and n are real number such that m2>n>0 then show that the equation ax2+2mbx+nc=0 has real roots.

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Solution

Given roots of the equation
ax2+2bx+c=0 are real.
(2b)24ac0
b2ac0 .............(1)

Discriminant of ax2+2mbx+nc=0 is
D=(2mb)24anc
D=4m2b24anc .............(2)

From (1) b2ac ........(3)
and given m2>n .......(4)
So, from (3) and (4),
b2m2anc
4m2b24anc0
D0 {from (2)}
Hence roots of equation ax2+2mbx+nc=0 are real.

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