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Question

Prove that the roots of the following equation are rational
(a+cb)x2+2cx+(b+ca=0)

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Solution

If discriminant is a perfect square then roots are rational.
D=b24ac
D=(2c)24(a+cb)(b+ca)
D=4c24[ab+aca2+bc+c2acb2bc+ab]
D=4c24[a2b2+c2+2ab]
D=4c24c2+4(a2+b22ab)
D=4(ab)2
D=(2(ab))2
D is perfect square of 2(ab)
Hence roots are rational.

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