Number of real solution (x,y) of the equation x2+1x2=21−y2 is
x2+1x2≥2 for all x (AM−GM inequality)
21−y2≥2
2y2≤1
But y2≥0 for real y
∴2y2≥20=1
∴2y2=1 i.e., y=0 is the only value;
if y=0,x2+1x2=2
∴x=±1
(1,0),(−1,0) are the 2 solutions.
Number of real solutions =2.
x5(x3−1)+(1−x3)x4=(x3−1)x4(x9−1)
=(x3−1)2(x6+x3+1)x4