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x1 is the real root of the equation x2+ax+b=0, x2 is the real root of the equation x2axb=0.

Prove that the equation x2+2ax+2b=0 has a real root, enclosed between x1 and x2. (a and b are real numbers).

We are given a polynomial P(x) and numbers a1, a2, a3, b1, b2, b3 such that a1a2a30. It turned out that P(a1x+b1)+P(a2x+b2)=P(a3x+b3) for any real x. Prove that P(x) has at least one real root.