First slide

Theory of equations

Question

If a + b + c = 0 and a, b, c are rational, then the roots of the equation (b + c – a) x² + (c + a – b) x + (a + b– c) = 0 are

Moderate

A

rational
B

irrational
C

imaginary
D

equal

Solution

We have,
$\begin{aligned} D = & (c + a - b)^{2} - 4 (b + c - a) (a + b - c) \\ = & (a + b + c - 2 b)^{2} - 4 (a + b + c - 2 a) (a + b + c - 2 c) \\ = & (- 2 b)^{2} - 4 (- 2 a) (- 2 c) = 4 (b^{2} - 4 ac) \\ = & 4 [(- a - c)^{2} - 4 ac] = 4 (a - c)^{2} \\ = & [2 (a - c)]^{2} = perfect square \end{aligned}$
$∴$ Roots are rational

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