If $a, b, c$ are rational, then the roots of $(a + b + c) x^{2} - 2 (a + c) x + (a - b + c) = 0$ are

real

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equal

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rational

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imaginary

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Solution

The correct options are
A rational
D real
Let $f (x) = (a + b + c) x^{2} - 2 (a + c) x + (a - b + c)$
$\Rightarrow f (1) = a + b + c - 2 a - 2 c + a - b + c$ $= 0$
$∴ x = 1$ is a root .
Product of roots $= \frac{C}{A} = \frac{a - b + c}{a + b + c}$
$∴$ Second root is $\frac{a - b + c}{a + b + c}$ ...Since, 1st root is $1$ .

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