Question

What is the value of (x-a)(x-b)(x-c)(x-d)….(x-z); if x=3

Answer 1

Answer:

We have,

(x−a)(x−c)+2(x−b)(x−d)=0

⇒3x

2

−(a+c+2b+2d)x+(ac+2bd)=0

Discriminant

={(a+2d)+(c+2b)}

2

−4.3(ac+2bd)

={(a+2d)+(c+2b)}

2

−12(ac+2bd)

={(a+2d)−(c+2b)}

2

−4(a+2d)(c+2b)−12(ac+2bd)

={(a+2d)−(c+2b)}

2

+4ac+8ab+8cd+16bd−12ac−24bd

={(a+2d)−(c+2b)}

2

+8(ab+cd−ac−bd)

={(a+2d)−(c+2b)}2+8(c−b)(d−a)>0

∵a<b<c<d⇒c−b>0and d−a>0

∴ The roots of the given equation are real and distinct