The nature of roots of the equation $(a + b + c) x^{2} - 2 (a + b) x + (a + b - c) = 0$ is

Rational

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Irrational

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Imaginary

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(A) & (B)

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Solution

The correct option is C Rational
The given equation is $(a + b + c) x^{2} - 2 (a + b) x + (a + b - c) = 0$
$D = B^{2} - 4 A C$
$D = 4 (a + b)^{2} - 4 (a + b + c) (a + b - c)$
$= 4 [(a + b)^{2} - (a + b)^{2} + c^{2}]$
$= 4 c^{2} = (2 c)^{2}$
$\Rightarrow D \geq 0$ and also a perfect square, hence roots are rational .

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