Question

If $a,b,c,d\in R$ such that $a<b<c<d$ then the
roots of the equation $\left(x-a\right)(x-c)+2(x-b)(x-d)=0$

• A. are imaginary
• B. are equal
• C. are real and distinct lying between a and b
• D. real and distinct between a and d

Answer 1

The correct option is B are equal

$a<b<c<d$

$(x-a)(x-c)+2(x-b)(x-d)=0$

$\Rightarrow 3x^2-(a+c+2b+2d)x+ac+2bd=0$

$D\equiv (a+c+2b+2d{)}^2-12(ac+2bd)$

$\equiv a^2+c^2+4b^2+4d^2+2ac+4cb+8bd+4da-12ac-24bd$

$\equiv a^2+c^2+4b^2+4d^2-10ac+4cb-18bd+4da<0$

$\Rightarrow$ Roots are imaginary.