#### Answer

$b=\{ -10,10 \}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To find the value of $
b
$ such that the given equation, $
p^2+bp+25=0
,$ will have $1$ rational solution, equate the discriminant to $0$ and solve for the variable.
$\bf{\text{Solution Details:}}$
In the equation above, $a=
1
,$ $b=
b
,$ and $c=
25
.$ Using the Discriminant Formula which is given by $b^2-4ac,$ the discriminant is
\begin{array}{l}\require{cancel}
(b)^2-4(1)(25)
\\\\=
b^2-100
.\end{array}
Equating the discriminant to $0$ so that the given equation will have $1$ rational solution, then
\begin{array}{l}\require{cancel}
b^2-100=0
\\\\
b^2=100
.\end{array}
Taking the square root of both sides, then
\begin{array}{l}\require{cancel}
b=\pm\sqrt{100}
\\\\
b=\pm10
.\end{array}
Hence, the given equation has $1$ rational solution when $
b=\{ -10,10 \}
.$