Stop and Think!
If the above quadratic equation has distinct real roots in for real numbers and with non-zero , then what can you say about the sign of and ?
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Say α , β are the roots of the given equation. Now considering, 5 a + 2 b + c . Since a = 0 , therefore, it is legitimate to divide this expression by a . Dividing 5 a + 2 b + c by a yields 5 − 2 ( − a b ) + a c . Since a − b = α + β and a c = α ⋅ β , therefore we have 5 − 2 ( α + β ) + α ⋅ β . After some simple manipulations, 5 − 2 ( α + β ) + α ⋅ β can be written as ( α − 2 ) ( β − 2 ) + 1 . Since α , β ϵ ( 1 , 2 ) , therefore, ( α − 2 ) ( β − 2 ) > 0 . Consequently, ( α − 2 ) ( β − 2 ) + 1 > 1 . So, a 5 a + 2 b + c > 1 . Clearly, this is possible only when 5 a + 2 b + c and a have the same sign.