## Precalculus (6th Edition)

$b=30$ or $b=-30$
RECALL: There are two forms of perfect square trinomials: (1) $m^2+2mn+n^2$, which is the square of $(m+n)^2$; and (2) $m^2-2mn+n^2$, which is the square of $(m-n)^2$ The given trinomial has: $m^2 = 9p^2=(3p)^2$, which means that $m=3p$ $n^2=25=5^2$, which means that $n=5$ Thus, the given trinomial will be a perfect square if (i) $bp=2mn=2(3p)(5) = 30p$, which means that $b=30$; and when (ii) $bp=-2mn=-2(3p)(5) = -30p$, which means that $b=-30$ Therefore, the given polynomial is a perfect square when $b=30$ and when $b=-30$