# If a+b+c=0 and a, b, care rational, prove that the roots of the equation (b+c-a) + (c+a-b)x + (a+b-c)=0 are rational.

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by Nitya8

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by Nitya8

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Putting x=1, we see that the equation is satisfied. So 1 is a root of the equation. This is verification.
In fact, you can find the conditions when 1 will be a root of the general quadratic equation Ax2+Bx+C=0, by,
x=−B±B2−4AC√2A=1,
B2−4AC=(2A+B)2,
B2−4AC=4A2+B2+4AB,
4A(A+B+C)=0,
A+B+C=0,
where the last equality comes from the fact that in a quadratic equation, A is non-zero.
So 1 is a root if and only if the sum of the coefficients is 0. (Again this can be verified by putting x=1 in the general equation directly.)