Question

CMR nếu a=b+c thì a^4+b^4+c^4=2a^2b^2+2b^2c^2+2c^2a^2

Answer 1

a + b = c => (a + b)² = c² <=> a²+ b² + 2ab = c² 
=> c^4 = (a² + b² + 2ab)² 
=> c^4 = a^4 + b^4 + 6a²b² + 4a^3.b + 4a.b^3 

vậy: a^4 + b^4 + c^4 = 2a^4 + 2b^4 + 6a²b² + 4a^3.b + 4a.b^3 
= 2a^4 + 2a²b² + 4a^3.b + 2b^4 + 2a²b² + 4a.b^3 + 2a²b² 
= 2a²(a² + b² + 2ab) + 2b²(b² + a² + 2ab) + 2a²b² 
= 2a²(a + b)² + 2b²(a + b)² + 2a²b² 
= 2a²b² + 2(a + b)²(a² + b²) 
= 2a²b² + 2c²(a² +b²) 
= 2a²b² + 2b²c² + 2c²a² (đpcm)