Both the roots of the equation are always
- A.positive
- B.negative
- C.real
- D.None of the above
Show answer and solution
Correct option: C
Solution
The discriminant of this quadratic can be found by computing it directly. Expanding, the equation is 3x² - 2(a+b+c)x + (ab+bc+ca) = 0.
The discriminant is 4(a+b+c)² - 12(ab+bc+ca) = 4[(a+b+c)² - 3(ab+bc+ca)] = 4[a² + b² + c² - ab - bc - ca] = 2[(a-b)² + (b-c)² + (c-a)²].
This is always ≥ 0 for all real a, b, c, so the roots are always real. (They are equal only when a = b = c.)
The discriminant is 4(a+b+c)² - 12(ab+bc+ca) = 4[(a+b+c)² - 3(ab+bc+ca)] = 4[a² + b² + c² - ab - bc - ca] = 2[(a-b)² + (b-c)² + (c-a)²].
This is always ≥ 0 for all real a, b, c, so the roots are always real. (They are equal only when a = b = c.)
[1980]