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The quadratic equation (x-a)(x-b)+(x-b)(...

The quadratic equation `(x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0` has equal roots if

A

positive

B

negative

C

real

D

None of the above

Text Solution

Verified by Experts

The correct Answer is:
C
`(x-a)(x-b)+(x-b)(x-c)(x-a)=0`
`implies 3x^(2)-2(a+b+c)x+(ab+bc+ca)=0`
Now, discriminant `=4 (a+b+c)^(2)-12(ab+bc+ca)`
`=4{a^(2)+b^(2)+c^(2)-ab-bc-ca}`
`=2{(a-b)^(2)+(b-c)^(2)+(c-a)^(2)}`
which is always positive.
Hence, both roots are real.
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