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Olympiad workbook

Exam: Olympiad workbook
Subject: IOQM
Chapter: Quadratic Equation

Olympiad workbook · IOQM · Quadratic Equation

Suppose a, b are positive real numbers such that a a +b b =183, a b...

Suppose

a, b

are positive real numbers such that

a \sqrt{a}+b \sqrt{b}=183, a \sqrt{b}+b \sqrt{a}=182

. Find

\frac{9}{5}(a+b)

.

Correct answer

73

Step-by-step solution

a \sqrt{a}+b \sqrt{b}=183

and

a \sqrt{b}+b \sqrt{a}=182

we have to find

\frac{9}{5}(a+b)

Let

a=A^2, b=B^2

\begin{aligned} & A^3+B^3=183 \ldots \ldots \ldots \ldots . \text { (i) } \\ & A^2 B+B^2 A=182 \ldots \ldots \ldots . . \text { (ii) } \\ & (1)+(2) \\ & A^3+B^3+3 A B(A+B)=183+3 \times 182 \\ & (A+B)^3=183+546 \\ & (A+B)^3=729 \\ & \Rightarrow A+B=9 \end{aligned}

\begin{aligned} & \text { from }(2) \rightarrow A B(A+B)=182 \\ & A B=\frac{182}{9} \\ & a+b=A^2+B^2=(A+B)^2-2 A B \\ & =81-\frac{364}{9}=\frac{365}{9} \\ & \frac{9}{5} \times \frac{365}{9}=73\end{aligned}

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