Problem: Let F=66x2+16x+3 m be the square of an expression which is linear in x. Then m has a particular value between:
Answer Choices:
A. 3 and 4
B. 4 and 5
C. 5 and 6
D. -4 and -3
E. -6 and -5
Solution:
Method I. F=x2+38x+2m=(x+r)2=x2+2rx+r2
∴2r=38 and r2=2m,∴r=34 and m=932=395 and 3<395<4.
Comment. When m=932, F=(x+34)2.
Method II. By completing the square we find that the trinomial x2+38x+916= (x+34)2. Therefore, x2+38x+916=x2+38x+2m, so that 2m=916 and m=932.