If `a ,b ,c ,d in R ,` then the equation `(x^2+a x-3b)(x^2-c x+b)(x^2-dx+2b)=0` has a. 6 real roots b. at least 2 real roots c. 4 real roots d. none of these

Options

A. 6 real roots
B. at least 2 real roots
C. 4 real roots
D. 3 real roots

Answer

2

Explanation

The discriminants of the given equation are `D_(1) = a^(2) + 12b`,
`D_(2) = c^(2) - 4b and D_(3) = d^(2) - 8b.`
`therefore D_(1) + D_(2) + D_(3) = a^(2) + c^(2) + d^(2) ge 0`
Hence, at least one fo `D_(1), D_(2), D_(3)` is non-negative. Therefore, the equation has at least two real roots.