If the expression `x^2+2(a+b+c)x+3(bc+ca+ab)` is a perfect square then

Options

A. a = b = c
B. `a = pm b = pm c`
C. `a = b ne c`
D. none of these

Answer

1

Explanation

Given quadratic expression is `x^(2) + 2(a + b + c)x + 3(bc + ca + ab).`
This quadratic expression will be a perfect square if the discriminant of its corresponding equation is zero. Hence,
`4(a + b + c)^(2) - 4 xx 3(bc + ca + ab) = 0`
or `(a + b + c)^(2) - 3(bc + ca + ab) = 0`
or `a^(2) + b + c^(2) + 2ab + 2bc + 2ca - 3(bc + ca + ab) = 0`
or `a^(2) + b^(2) + c^(2) - ab bc - ca = 0`
or `1/2[2a^(2) + 2b^(2) + 2c^(2) - 2ab - 2bc - 2ca] = 0`
or `1/2[(a^(2) + b^(2) - 2ab) + (b^(2) + c^(2) - 2bc) + (c^(2) + a^(2) - 2ca)] = 0`
or `1/2[(a-b)^(2) + (b - c)^(2) + (c - a)^(2)] = 0`
which is possible only when `(a - b)^(2) = 0, (b - c)^(2) = 0`
and `(c - a)^(2) = 0,` i.e., a = b = c.