First slide

Theory of equations

Question

Both the roots of the given equation
$(x - a) (x - b) + (x - b) (x - c) + (x - c) (x - a) = 0$ are always

Moderate

A

Positive
B

Negative
C

Real
D

Imaginary

Solution

Given equation
$(x - a) (x - b) + (x - b) (x - c) + (x - c) (x - a) = 0$
can be re-written as
$3 x^{2} - 2 (a + b + c) x + (ab + bc + ca) = 0 ∆ = 4 {{(a + b + c)}^{2} - 3 (ab + bc + ca)} (∵ b^{2} - 4 ac = ∆) = 4 (a^{2} + b^{2} + c^{2} - ab - bc - ac) = 2 {{(a - b)}^{2} + {(b - c)}^{2} + {(c - a)}^{2}} \geq 0$
Hence both roots are always real.

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