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

• A. real and distinct
• B. non-real complex
• C. real and equal
• D. integers

Answer 1

The correct option is A real and distinct

Let $k(>0)$ be the common difference of A.P., then $b=a+k,c=a+2k,d=a+3k$

Equation is $3x^2-\left[\right(a+c)+2(b+d\left)\right]x+(ac+2bd)=0$

$\Rightarrow 3x^2-(6a+10k)x+a(a+2k)+2(a+k)(a+3k)=0$

Discriminant, $D=(6a+10k{)}^2-4\times 3\times (a^2+2ak+2a^2+8ak+6k^2)$
$=36a^2+100k^2+120ak-12(3a^2+10ak+6k^2$
$=28k^2>0[\because k\ne 0]$

So, the roots are real and distinct.