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標題:

A.maths

發問:

if a,b,c are real numbers and not all equal, prove that the quadratic equation (c-a)x^2-2(a-b)x+(b-c)=0................(*) has unequal real roots. x^2=x的2次方

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∵(a+b+c)^2=a^2+b^2+c^2+2ab+2ac+2bc ∴(a+b+c)^2 >= 0 ∴a^2+b^2+c^2+2ab+2ac+2bc >= 0--------(1) And also ∵ a^2+b^2+c^2 > 0 ∴ 2ab+2ac+2bc 0 and -(ab+bc+ac) > 0 (From (2) ) ∴Concluding that the discriminant >0 ∴(*) has unequal real roots

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