Question 22

Consider the equation $$(x-a)(x-b)=0$$ which has two identical roots. What should be the condition under which $$(x-a)(x-b)=c$$ has two distinct real roots ?

Solution

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(x-a)(x-b) = 0 having identical roots means a = b
Hence, the expression now effectively becomes (x-a)(x-a) = c

For x to have different values, we should have c > 0 because at c = 0, we will end up with x = a AND c < 0 is not possible for rational values of a and x. 
For c > 0, we will have x = a + (root c) and x = a - (root c) 

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