If α, β are roots of equation (x-a)(x-b) + (x-α)(x-β) = 0. Find roots of the equation (x-α)(x-β) = C

The problem states that a and b are the roots of the equation (x-a)(x-b) + (x-α)(x-β) = 0, which means substituting x = a and x = b must yield zero. Substituting x = a gives (a-a)(a-b) + (a-α)(a-β) = 0, simplifying to (a-α)(a-β) = 0, so the equation (x-α)(x-β) = C evaluates to zero when x = a. Similarly, substituting x = b gives (b-α)(b-β) = 0, meaning x = b is also a root of (x-α)(x-β) = C. Therefore, the roots of the equation (x-α)(x-β) = C are exactly a and b.

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