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(^378) Answers to Exercises and Solutions to Problems
Thus, from (b), if p < -l/2, there is no solution. If p < -l/4, then a
and b are not solutions. If p > 7/2, c and d are not solutions. If p < 712,
then d 5 -4, SO that 3d +p+4 < 0 and x = d denies condition (c). Hence,
d is never a solution of the given equation.
Suppose that c is a solution. Then, since -l/2 5 p, c _< -4 +2fi < -1,
so that, by condition (a), c 5 -2, and, by condition (c), p 2 -4 + 6 = 2.
Hence, if p < 2, then c is not a solution. Thus, there are no solutions if
p < -l/4.
Let p 1 -l/4. Then
2p+l-a2 = (Jxp+ l/q2
3a+p+4= (&v+3/2)2
a2+9a+3p+9 = p-a+9a+3p+9=8a+4p+9
= (&X+2)2
and a satisfies the equation.
Also
If p 1 2, then
2p+ 1 - b2 = (dpx- 1/2)2
3b+p+4= (dm-3/2)2
b2 + 9b + 3p + 9 = (,/4x - 2)2.
= ,/w-2= ,/b2+9b+3p+9.
If 0 < p < 2, then the left side is
(t/m - l/2) + (3/2 - dm) = 1,
which is not equal to
db2+9b+3p+9= I&7%21.
If -l/4 5 p 5 0, then the left side is
(l/2 - dm) + (3/2 - JpiT;I) = 2 - d&i
and the given equation is satisfied.
Summing up, we have the table:
Range ofp Is a b c d a solution?
p < -l/4 N N N N
-l/4 5 p 5 0 YYNN
o<p<2 YNNN
25P YY?N