Problem: x x
49 − x 2 − 25 − x 2 = 3 4 9 − x 2 − 2 5 − x 2 = 3
What is the value of 49 − x 2 + 25 − x 2 4 9 − x 2 + 2 5 − x 2
Answer Choices:
A. 8 8 33 + 3 3 3 + 3 9 9 2 10 + 4 2 1 0 + 4 12 1 2
Solution:
Let
a = 49 − x 2 − 25 − x 2 and b = 49 − x 2 + 25 − x 2 . a = 4 9 − x 2 − 2 5 − x 2 and b = 4 9 − x 2 + 2 5 − x 2 .
Then a b = ( 49 − x 2 ) − ( 25 − x 2 ) = 24 a b = ( 4 9 − x 2 ) − ( 2 5 − x 2 ) = 2 4 b = 24 a = 24 3 = 8 b = a 2 4 = 3 2 4 = 8
The given equation can be solved directly. Adding 25 − x 2 2 5 − x 2 15 = 6 25 − x 2 1 5 = 6 2 5 − x 2 x 2 x 2 x 2 = 75 4 x 2 = 4 7 5 49 − x 2 + 25 − x 2 4 9 − x 2 + 2 5 − x 2 ( A ) 8 ( A ) 8
The problems on this page are the property of the MAA's American Mathematics Competitions