Question

Let x and y be positive real numbers. Define a = 1 + x/y and b = 1 + y/x. if a^2 +b^2 =15, compute a^3 + b^3

Answer 1

Answer:

50

Step-by-step explanation:

Not very tough question. Apply simple mathematics

Answer 2

Solution :

It is defined that for x and y being positive real numbers ;

• a = 1 + x/y

• b = 1 + y/x

• a² + b² = 15

We have to find the value of a³ + b³

a = 1 + x/y

=> ( a - 1) = x/y ..... [ 1 ]

b = 1 + y/x

=> ( b - 1) = y/x ....... [ 2 ]

Multiplying [ 1 ] by [ 2 ]

=> ( a - 1)( b - 1) = 1

=> ab - a - b + 1 = 1

=> ab - a - b = 0

=> ab = a + b

Let ab = a + b =. t

a² + b² = 15

=> ( a + b)² - 2ab = 15

=> t² - 2t = 15

=> t² - 2t - 15 = 0

=> t² - 5t + 3t - 15 = 0

=> t ( t - 5) + 3( t - 5) = 0

=> ( t + 3)( t - 5) = 0

=> t = -3 or 5

t cant be negative.

So, the only possible value of t is 5

a³ + b³

=> ( a + b )³ - 3ab ( a + b )

=> t³ - 3t²

=> 125 - 75

=> 50

This is the required answer .

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Note : To solve this question , you can use either of the methods used by the first answerer and me. The main concept to be applied in these types of problems is that it should get easily simplified in any method and you will know you are solving the right way . Some of the methods are longer than others ; best of luck trying :P