Theory of Equations 5 Question 17
17. 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.
(1984, 1M)
Analytical & Descriptive Question
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Solution:
- Let $f(x)=(x-a)(x-c)+2(x-b)(x-d)$
Here,
$$ \begin{aligned} & f(a)=+ve \\ & f(b)=-ve \\ & f(c)=-ve \\ & f(d)=+ve \end{aligned} $$
$\therefore$ There exists two real and distinct roots one in the interval $(a, b)$ and other in $(c, d)$.
Hence, statement is true.
