Math Olympiad | A Nice Algebra Problem | VIJAY Maths 

SENSATIONAL!

(a + b)² = a² + 2ab + b² → given: a + b = 1 1 = a² + 2ab + b² → given: a² + b² = 2 1 = 2 + 2ab ab = - 1/2 (a + b)⁴ = (a + b)².(a + b)² (a + b)⁴ = (a² + 2ab + b²).(a² + 2ab + b²) (a + b)⁴ = a⁴ + 2a³b + a²b² + 2a³b + 4a²b² + 2ab³ + a²b² + 2ab³ + b⁴ (a + b)⁴ = a⁴ + b⁴ + 6a²b² + 4a³b + 4ab³ (a + b)⁴ = a⁴ + b⁴ + 6a²b² + 4ab.(a² + b²) → given: a² + b² = 2 (a + b)⁴ = a⁴ + b⁴ + 6a²b² + 8ab (a + b)⁴ = a⁴ + b⁴ + ab.(6ab + 8) → recall: ab = - 1/2 (a + b)⁴ = a⁴ + b⁴ + (- 1/2).(- 3 + 8) (a + b)⁴ = a⁴ + b⁴ - (5/2) ← equation (1) a⁴ + b⁴ = (a + b)⁴ + (5/2) → given: a + b = 1 a⁴ + b⁴ = 1 + (5/2) a⁴ + b⁴ = 7/2 ← equation (2) (a + b)⁸ = [(a + b)⁴]² (a + b)⁸ = [a⁴ + b⁴ - (5/2)]² (a + b)⁸ = a⁸ + a⁴b⁴ - (5/2).a⁴ + a⁴b⁴ + b⁸ - (5/2).b⁴ - (5/2).a⁴ - (5/2).b⁴ + (25/4) (a + b)⁸ = a⁸ + b⁸ + 2a⁴b⁴ - 5a⁴ - 5b⁴ + (25/4) (a + b)⁸ = a⁸ + b⁸ + 2a⁴b⁴ - 5.(a⁴ + b⁴) + (25/4) → recall: ab = - 1/2 (a + b)⁸ = a⁸ + b⁸ + 2.(- 1/2)⁴ - 5.(a⁴ + b⁴) + (25/4) (a + b)⁸ = a⁸ + b⁸ - 5.(a⁴ + b⁴) + (51/8) → recall (2): a⁴ + b⁴ = 7/2 (a + b)⁸ = a⁸ + b⁸ - (35/2) + (51/8) (a + b)⁸ = a⁸ + b⁸ - (89/8) ← equation (3) a⁸ + b⁸ = (a + b)⁸ + (89/8) → given: a + b = 1 a⁸ + b⁸ = 1 + (89/8) a⁸ + b⁸ = 97/8 ← equation (4) (a + b)¹² = (a + b)⁸. (a + b)⁴ → recall (3) (a + b)¹² = [a⁸ + b⁸ - (89/8)].(a + b)⁴ → recall (1) (a + b)¹² = [a⁸ + b⁸ - (89/8)].[a⁴ + b⁴ - (5/2)] (a + b)¹² = a¹² + a⁸b⁴ - (5/2).a⁸ + a⁴b⁸ + b¹² - (5/2).b⁸ - (89/8).a⁴ - (89/8).b⁴ + (445/16) (a + b)¹² = a¹² + b¹² + a⁸b⁴ + a⁴b⁸ - (5/2).a⁸ - (5/2).b⁸ - (89/8).a⁴ - (89/8).b⁴ + (445/16) (a + b)¹² = a¹² + b¹² + a⁴b⁴.[a⁴ + b⁴] - (5/2).[a⁸ + b⁸] - (89/8).[a⁴ + b⁴] + (445/16) (a + b)¹² = a¹² + b¹² + [a⁴ + b⁴].[a⁴b⁴ - (89/8)] - (5/2).[a⁸ + b⁸] + (445/16) → recall (2): a⁴ + b⁴ = 7/2 (a + b)¹² = a¹² + b¹² + [7/2].[a⁴b⁴ - (89/8)] - (5/2).[a⁸ + b⁸] + (445/16) → recall (4): a⁸ + b⁸ = 97/8 (a + b)¹² = a¹² + b¹² + [7/2].[a⁴b⁴ - (89/8)] - (5/2).[97/8] + (445/16) → recall: ab = - 1/2 (a + b)¹² = a¹² + b¹² + [7/2].[(- 1/2)⁴ - (89/8)] - (5/2).[97/8] + (445/16) (a + b)¹² = a¹² + b¹² + [7/2].[- (177/16)] - (485/16) + (445/16) (a + b)¹² = a¹² + b¹² - (1239/32) - (485/16) + (445/16) (a + b)¹² = a¹² + b¹² - (1319/32) a¹² + b¹² = (a + b)¹² + (1319/32) → given: a + b = 1 a¹² + b¹² = 1¹² + (1319/32) a¹² + b¹² = 1351/32