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Question

The equation (xx+1)2+(xx1)2=a(a1) has
  1. four real roots if a>2
  2. four real roots if a<1
  3. two real roots if 1<a<2
  4. no real roots if a<1


Solution

The correct options are
A four real roots if a>2
B four real roots if a<1
C two real roots if 1<a<2
(xx+1)2+(xx1)2=a(a1)(xx+1+xx1)22(xx+1)(xx1)=a(a1)(2x2x21)2(2x2x21)a(a1)=0

Let 2x2x21=t
t2ta(a1)=0t=a or t=1a2x2x21=a or 2x2x21=1ax=±aa2 or x=±a1a+1

When a<1 all roots are real
1<a<2x=±a2ai,±a1a+1 two real roots
When a>2 all roots are real
 

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