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  4. If a < b < c < d, then the roots of the equation (x - a) (x - c) + 2 (x - b) (x - d) = 0 are real and distinct.

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Q. If $a < b < c < d$, then the roots of the equation $(x - a) (x - c) + 2 (x - b) (x - d) = 0$ are real and distinct.

IIT JEEIIT JEE 1984Complex Numbers and Quadratic Equations

Solution:

Let $ f (x ) = (x - a) (x - c) + 2 (x - b) (x - d)$
Here, $f (a) =$ + ve
$f (b) = $- ve
$f (c) =$ - ve
$f (d) =$ + ve
$\therefore $ There exists two real and distinct roots one in the interval $(a, b)$ and other in $(c, d)$.
Hence, statement is true.