Identity of Quadratic equations:

If the quadratic equation is fulfilled by more than two distinct numbers complex or real, then it becomes an identity i.e. a = b = c = 0. For example

 is fulfilled by three value of x which are a, b and c. therefore this is an identity in x.

Illustration:         Suppose p(x) =.b²

                                    a ≠ b ≠ c. Show that p(x) has the property that p(y) = y² for all y ∈ R

Solution:                   Note that P(a) = a², P(b) = b² and P(c) = c².

                                    Suppose the polynomial Q(x) = P(x) - x², Q(x) has degree atmost 2.

                                    Also Q(a) = Q(b) = Q(c) = 0 => Q(x) has 3 distinct roots.

                                    It follows that Q(x) is normally zero

                                    i.e. Q(y) = 0 ∀ y ∈ R

                                    =>        P(y) - y² = 0 ∀ y ∈ R => P(y) = y² ∀y ∈ R

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