We have that
pq = p + q (1)
p - q = 7 ⇒ p = 7 + q (2)
Sub (2) into (1) and we have that
(7 + q)q = 7 + 2q simplify
7q + q^2 = 7 + 2q rearrange
q^2 + 5q - 7 = 0 complete the square on q
q^2 + 5q + (25/4) = 7 + 25/4
(q + 5/2)^2 = 53/4 take the positive root
q + 5/2 = √53/2
q = √53/2 - 5/2 = [√53 - 5 ] / 2
q^2 = [ 53 - 10√53 + 25] / 4 = [ 78 - 10√53 ] / 4 = [39 - 5√53] / 2
So....p = √53/2 - 5/2 + 7 = √53/2 + 9/2 = [ √53 + 9 ] / 2
p^2 = [ 53 + 18√53 + 81] / 4 = [ 134 +18√53 ] / 4 = [ 67 + 9√53 ] / 2
Simplifying the given expression we have that
(pq)^2 (p + q)^2
_________ = ________
p^2 + q^2 p^2 + q^2
p + q = √53 + 2
(p+ q)^2 = ( 53 + 4√53 + 4) = (57 + 4√53)
p^2 + q^2 = [106 + 4√53 ] / 2 = 53 + 2√53
So we have
(57 + 4√53 ) (53 - 2√53) 2597 + 98√53
___________ ___________ = ______________
( 53 + 2√53) (53 - 2√53) 2597
So
a + b + c + d =
2597 + 98 + 53 + 2597 = 5345