olympiad math question | find the value of x

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[@rghfi67y4f8y]

missed 2nd root, if t=-5 x=1/12

[@skill_issuesmo7367]

t cant be negative

[@maxence_lvl8331]

Hi! I don't quite understand... I achieved to isolate the x's with a polynomial which equals to 0 (--> 48x^2-16x+1=0). And i found two roots : 1/4 and 1/12. Of course 1/12 doesn't work, but I don't understand why. Could you explain this to me please ? Thanks

[@lightyagami1752]

Substitute it back into the original equation and you'll find it does not satisfy it. This is a known phenomenon when you take squares or other even higher powers of equations, you may introduce redundant or extraneous roots. This is because by standard convention, sqrt(x) with the usual symbol, only refers to the non-negative square root. But when you square you unwittingly introduce the possibility of including the negative square root. That's why you need to check every single solution by substitution into the original equation.

[@maxence_lvl8331]

@@lightyagami1752 Thanks for the response!

[@tejasvadhyani7860]

@@lightyagami1752 I think 1/12 is a valid value since sqrt(25) = -5 as well. Since the degree is 2, we would get 2 zeroes(values)