First slide

Theory of equations

Question

If the roots of the equation $(x - b) (x - c) + (x - c) (x - a) + (x - a) (x - b) = 0$ are equal then

Moderate

A

$a + b + c = 0$
B

$a + b ω + c ω^{2} = 0$
C

$a - b + c = 0$
D

none of these

Solution

The given equation can be written as
$3 x^{2} - 2 (a + b + c) x + (a b + b c + c a) = 0$
The roots of this equation will be equal if
$\begin{array}{l} 4 (a + b + c)^{2} - 12 (a b + b c + c a) = 0 \\ \Rightarrow a^{2} + b^{2} + c^{2} - a b - b c - c a = 0 \\ \Rightarrow (a + b w + c w^{2}) (a + b w^{2} + c w) = 0 \\ \Rightarrow a + b ω + c ω^{2} = 0 or, a + b ω^{2} + c ω = 0 \end{array}$

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