If a < b < c < d, then the nature of roots of ( x – a) (x – c) + 2 (x – b) (x – d) = 0 is
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Question Details
Mathematics
Quadratic Equation
Engineering Entrance Exam
Question
If a < b < c < d, then the nature of roots of ( x – a) (x – c) + 2 (x – b) (x – d) = 0 is
Solution
Here, \(3{x^2} - (a + c + 2b + 2d)x + (ac + 2bd) = 0\)
\(\therefore D = {(a + c + 2b + 2d)^2} - 12(ac + 2bd)\)
\( = {\left[ {(a + 2d) - (c + 2b)} \right]^2} + 4(a + 2d)(c + 2b) - 12(ac + 2bd)\)
\( = {\left[ {(a + 2d) - (c + 2b)} \right]^2} + 8(c - b)(d - a) > 0\)
Hence roots are real and unequal