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You have, (7-4√3)(7+4√3)=1. You can see that for x=1 equation is satisfied, but too sad need x<0. Then, for convenience, put x=-y where y>0. You have, [(7+4√3)^{y}+(7-4√3)^{y}]=14[(7-4√3)(7+4√3)]^{y}. Which again falls to original equation but replacing x with y. Now y=1 hence x becomes -1. Answer x=-1
Most of the people could see that it satisfies for ±1 just by look. But to provide an understandable answer, I took me to write this thing