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​.