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

real and distinct
B

non-real complex
C

real and equal
D

integers

Solution

Let $λ$ be the common difference of the increasing A.P. Then,
$b = a + λ, c = a + 2 λ$ and $d = a + 3 λ,$ where $λ > 0 .$
$∴ (x - a) (x - c) + 2 (x - b) (x - d) = 0$
$\Rightarrow 3 x^{2} - 2 (3 a + 5 λ) x + a (a + 2 λ) + 2 (a + λ) (a + 3 λ) = 0$
Let $D$ be its discriminant. Then,
$\begin{array}{l} D = 4 (3 a + 5 λ)^{2} - 12 a (a + 2 λ) - 24 (a + λ) (a + 3 λ) \\ \Rightarrow D = 28 λ^{2} > 0 . \end{array}$
Hence, the roots of the given equation are real and distinct.

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App