Question 468457
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I presume you mean "how <i>two different</i> quadratic equations can have the same solution"


Consider the quadratic equation 


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ ax^2\ +\ bx\ +\ c\ =\ 0]


which has factors:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ (x\ -\ \phi_1)(x\ -\ \phi_2)\ =\ 0]


and therefore has the solutions *[tex \Large \phi_1] and *[tex \Large \phi_2]


Now consider:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ kax^2\ +\ kbx\ +\ kc\ =\ 0]


which has factors:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ k(x\ -\ \phi_1)(x\ -\ \phi_2)\ =\ 0]


and solutions *[tex \Large \phi_1] and *[tex \Large \phi_2]


hence there are infinite quadratic equations with a pair of given solutions *[tex \Large \phi_1] and *[tex \Large \phi_2] where *[tex \Large k(x\ -\ \phi_1)(x\ -\ \phi_2)\ =\ kax^2\ +\ kbx\ +\ kc\ =\ 0], one for every real number *[tex \Large k]


John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
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