Problem:
How many integers x satisfy the equation
(x2βxβ1)x+2=1?
Answer Choices:
A. 2
B. 3
C. 4
D. 5
E. none of these
Solution:
There are 3 ways ab can equal 1 when a and b are integers:
I) a=1;
II) a=β1,b even;
III) b=0,aξ =0.
In this problem a=x2βxβ1,b=x+2. We solve each case in turn.
Case I: x2βxβ1=1,
(xβ2)(x+1)=0,
x=2 or x=β1.
Case II: x2βxβ1=β1 and x+2 is even,
βx2βx=0x=0 or 1β
The choice x=0 is a solution, since then x+2 is even. However, x=1 is not a solution, since then x+2 would be odd.
Case III: x+2=0 and x2βxβ1ξ =0, x=β2.
This is a solution, since (β2)2β(β2)β1=5ξ =0.
Thus there are 4 solutions in all.