Question

if a<b<c<d, then the equation 3(x-a)(x-c) +5(x-b)(x-d) =0 has what type of root.​

Answer 1

We have,

(x−a)(x−c)+2(x−b)(x−d)=0

⇒3x2-(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 are real and distinct

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