Playing with real numbers

Number Theory Level 2

For real numbers x , y x,y and z z we have, x 2 + y 2 + 1 = 2 x x^2 + y^2 + 1 = 2x Then find the value of x 3 + y 5 x^3 + y^5 .

6 0 3 2 1

Manish Mayank by Manish Mayank , 20, India

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1 solution

Manish Mayank
May 28, 2015

We have, x 2 + y 2 + 1 = 2 x x^2 + y^2 + 1 = 2x = x 2 − 2 x + 1 + y 2 = 0 = x^2 - 2x + 1 + y^2 = 0 = ( x − 1 ) 2 + y 2 = 0 = (x-1)^{2} + y^2 = 0 We know that sum of squares of real numbers is 0 only when individual numbers are 0. So we get x = 1 x=1 and y = 0 y=0 Thus x 3 + y 5 = 1 + 0 = 1 \boxed {x^3 + y^5 = 1+0 = 1}

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