Find the value of 'a' so that x²-11x+a=0 and x²-14x+2a=0 have a common root.

Answer 1

We assume a≠0, because then it's clear that x=0 is a common root. So, we have

x²-11x+a: x = (11±√(121-4a))/2
x²-14x+2a: x = (14±√(196-8a))/2

we need some of those expressions to be equal. Some playing around shows that if

a = 24: 11+√(121-4a) = 14+√(196-8a)

x^2-11x+24 = (x-3)(x-8)
x^2-14x+48 = (x-6)(x-8)