(1)

The given equation will be a perfect square, if either the equations

have both roots common or each has equal roots.

Thus, we have the following cases:

CASE I When two equations have both roots common

In this case, we have

and

and .

CASE II When two equations have equal roots

and

and

and

and

There is no value of a satisfying these two equations simultaneously.

Hence, for , the given expression is a perfect square.

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