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If the quadratic equation (x - a)(x - b) + (x - b)(x - c) + (x - c)(x - a) = 0 has equal roots, then the value of a² + b² + c² is:

1. a + b + c
2. 2a + b + c
3. 3abc
4. ab + bc + ca

Correct answer: ab + bc + ca

Solution

Expanding gives 3x^2-2(a+b+c)x+(ab+bc+ca)=0. Equal roots need discriminant 0: 4(a+b+c)^2-12(ab+bc+ca)=0, so (a+b+c)^2=3(ab+bc+ca). Expanding, a^2+b^2+c^2+2(ab+bc+ca)=3(ab+bc+ca), hence a^2+b^2+c^2=ab+bc+ca.

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