Math Olympiad Algebra Problem | IMO Exam | Find the Value of x and y ?

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Math Olympiad Algebra Problem | IMO Exam | Find the Value of x and y ?

[walterwen]

Math Olympiad Algebra Problem: x⁴ + y⁴ = 97, x + y = 5; x, y = ?
(x + y)² = x² + y² + 2xy = 5² = 25; x² + y² = 25 – 2xy
(x + y)⁴ = (x² + y²)² = x⁴ + y⁴ + 2(xy)² = (25 – 2xy)²
97 + 2(xy)² = 625 – 100xy + 4(xy)², 2(xy)² – 100xy + 528 = 0
(xy)² – 50xy + 264 = (xy – 6)(xy – 44) = 0; xy – 6 = 0 or xy – 44 = 0
xy = 6, y = 5 – x, x(5 – x) = 6, x² – 5x + 6 = (x – 2)(x – 3) = 0
x – 2 = 0; x = 2, y = 5 – x = 3 or x – 3 = 0; x = 3, y = 5 – 3 = 2
xy = 44, y = 5 – x, x(5 – x) = 44, x² – 5x + 44 = 0, x = (5 ± i√151)/2
y = 5 – x = 5 – (5 ± i√151)/2 = (5 –/+ i√151)/2
Answer check:
x = 2, y = 3 or x = 3, y = 2, xy = 6: x + y = 2 + 3 = 3 + 2 = 5; Confirmed
x⁴ + y⁴ = (25 – 2xy)² – 2(xy)² = (25 – 12)² – 2(6²) = 97; Confirmed
x = (5 ± i√151)/2, y = (5 –/+ i√151)/2, xy = 44: x + y = 10/2 = 5; Confirmed
x⁴ + y⁴ = (25 – 88)² – 2(44²) = 63² – 2(1936) = 97; Confirmed
Final answer:
x = 2, y = 3; x = 3, y = 2
Two complex roots; x = (5 + i√151)/2, y = (5 – i√151)/2
or x = (5 – i√151)/2, y = (5 + i√151)/2

[E.h.a.b]

x^4 + y^4 = 97 ----> [1]
x + y = 5 ----> [2]

(x + y)^4 = 5^4
x^4 + 4 x^3 y + 6 x^2 y^2 + 4 x y^3 + y^4 = 625
(x^4 + y^4) + 4 x y( x^2 + y^2 ) + 6 x^2 y^2 = 625
(x^4 + y^4) + 4 x y( (x+y)^2 -2 x y) + 6 x^2 y^2 = 625
97 + 4 x y(25 -2 x y) + 6 (x y)^2 = 625
let x y = u
97 -625 + 4 u(25 -2 u) + 6 u^22 = 0
-528 + 100 u - 8 u^2 + 6 u^2 =0
2 u^2 - 100 u + 528 =0
u^2 - 50 u + 264 =0
u = ( 50 +/- √(2500 -4 (1)(264) )/2
u = 25 +/- √(361)
u = 25 +/- 19
x y = 25 +/- 19 ----> [3]
(x-y)^2 = (x+y)^2 - 4 x y
(x-y)^2 = 25 - 4 (25 +/- 19)
x-y = +/- √(25 -4(25 +/- 19)) ----> [4]
From [2] and [4]
2 x = 5 +/- √(25 -4(25 +/- 19))
x = (5 +/- √(25 -4(25 +/- 19)))/2 and y = 5 - x
x = {3, 2, (5+i√151)/2, (5-i√151)/2 }
y = {2, 3, (5-i√151)/2, (5+i√151)/2}
x, y = { (2, 3), (3, 2), ((5+i√151)/2, (5-i√151)/2), ((5-i√151)/2, (5+i√151)/2) }