Olympiad workbookIOQMQuadratic Equation

Suppose a, b are positive real numbers such that a a +b b =183, a b +b a =182 . Find 9 5 (a+b) .

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a a +b b =183 and a b +b a =182 we have to find 9 5 (a+b) Let a=A^2, b=B^2 aligned & A^3+B^3=183 . (i) & A^2 B+B^2 A=182 . . (ii) & (1)+(2) & A^3+B^3+3 A B(A+B)=183+3 182 & (A+B)^3=183+546 & (A+B)^3=729 & A+B=9 aligned aligned & from (2) A B(A+B)=182 & A B= 182 9 & a+b=A^2+B^2=(A+B)^2-2 A B & =81- 364 9 = 365 9 & 9 5 365 9 =73 aligned

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