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If a and b are nonzero unequal real numbers and (a−b)/a = b/(a−b), what is the sum of all possible

values for a/b?

 

I had no idea where to start with this one.

 Nov 9, 2022
 #2
avatar+118135 
+3

 

\(\frac{(a−b)}{a} =\frac{ b}{(a−b)}\\ (a-b)^2=ab\\ a^2-2ab+b^2=ab\\ a^2-3ab+b^2=0\\ a^2+(-3b)a+(b^2)=0\\ a=\frac{3b\pm\sqrt{9b^2-4b^2}}{2}\\ a=\frac{3b\pm b\sqrt{5}}{2}\\ \)

 

\(a=\frac{3b\pm b\sqrt{5}}{2}\\ \frac{a}{b}=\frac{3\pm \sqrt{5}}{2}\\ \text{if you add those two possibilities the sum is } 6/2 = 3 \)

 

 

You neeed to check this carefully.

 Nov 9, 2022

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