First slide

Theory of equations

Question

$If a, b, c, d are four consecutive terms of an increasing A.P. then the roots of the equation (x - a) (x - c) + 2 (x - b)$ $(x - d) = 0 are$

Moderate

A

non-real complex
B

real and equal
C

integer
D

real and distinct

Solution

$Let a = 1, b = 2, c = 3, d = 4$
given equation becomes
$\begin{array}{l} (x - 1) (x - 3) + 2 (x - 2) (x - 4) = 0 \\ \Rightarrow 3 x^{2} - 16 x + 19 = 0 \\ Now D = (- 16)^{2} - 4 .3.19 \\ = 256 - 228 \\ = 28 > 0 \end{array}$
Hence roots are real and distinct.

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