First slide

Theory of equations

Question

If $p, q, r$ are real and $p \neq q$ then the roots of the equation $(p - q) x^{2} + 5 (p + q) x - 2 (p - q) = r$ are

Moderate

A

real and equal
B

unequal and rational
C

uneaual and irrational
D

nothing can be said

Solution

The discriminant D of the given equation is
$D = 25 (p + q)^{2} + 8 (p - q)^{2} + 4 r (p - q)$
Clearly, its sign cannot be determined. So, nothing can be said about the nature of roots.

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