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If `a and b` are positive numbers and eah of the equations `x^2+a x+2b=0 and x^2+2b x+a=0` has real roots, then the smallest possible value of `(a+b)` is_________.

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Verified by Experts

The correct Answer is:
6
`a^(2)ge8band4b^(2)ge4a`
Now `b^(2)gea`
`impliesb^(4)gea^(2)ge8b(agt0,bgt0)`
`:.b^(3)ge8impliesbge2" "(1)`
Again `a^(2)ge8bandbge2`
`impliesa^(2)ge16`
or `age4" "(2)`
From (1) and (2),`(a+b)_("least")=6`.
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