Quadratic EquationHard
Question
If $a,b,c,p,q,r$ are non-zero real numbers, such that $a < b < c$ and $f(x) = (x - a)(x - b)(x - c) - p^{2}(x - a) - q^{2}(x - b) - r^{2}(x - c)$, then $f(x) = 0$ must have
Options
A.exactly 1 real root
B.exactly 3 distinct real roots
C.2 equal and 1 distinct real roots
D.nothing can be said
Solution
$f(a) = - q^{2}(a - b) - r^{2}(a - c) > 0$
$$f(c) = - p^{2}(c - a) - q^{2}(c - b) < 0 $$
$f(x) = 0$ has 3 distinct real roots.
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