First slide

Theory of equations

Question

If $a, b, c$ are non-zero rational numbers
such that $a + b + c = 0,$ then the roots of the equation
$(b + c - a) x^{2} + (c + a - b) x + (a + b - c) = 0$
are

Moderate

A

both rational
B

both irrational
C

both purely imaginary
D

one rational and one irrational

Solution

As $a + b + c = 0,$ we can write the equation
$2 a x^{2} + 2 b x + 2 c = 0$
or
$a x^{2} + b x + c = 0$
Note that $x = 1$ satisfies this equation and its other root is $c / a .$

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