Toán Lớp 8: Cho a, b thỏa mãn
a^3 − b^3 = b − a. Chứng minh rằng a = b.
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Toán Lớp 8: Cho a, b thỏa mãn
a^3 − b^3 = b − a. Chứng minh rằng a = b.
Comments ( 2 )
a3−b3=b−a↔(a−b)(a2+ab+b2)+(a−b)=0a3−b3=b−a↔(a−b)(a2+ab+b2)+(a−b)=0
↔(a−b)(a2+ab+b2+1)=0↔a=b↔(a−b)(a2+ab+b2+1)=0↔a=b
bởi vìa2+ab+b2+1=a2+2.a.b2+(b2)2−(b2)2+b2+1=(a+b2)2+3b24+1>0.
a^3 − b^3 = b − a ↔ (a − b)(a^2 + ab + b^2) + (a − b) = 0
↔ (a − b)(a^2 + ab + b^2 + 1) = 0 ↔ a = b
bởi vì a^2 + ab + b^2 + 1 = a^2 + 2.a. b/2+ (b/2)^2 − (b/2)^2 + b^2 + 1 =(a + b/2)^2 + {3b^2}/4+ 1 > 0.