## Intermediate Algebra (12th Edition)

$a=16$
$\bf{\text{Solution Outline:}}$ To find the value of $a$ such that the given equation, $am^2+8m+1=0 ,$ will have $1$ rational solution, equate the discriminant to $0$ and solve for the variable. $\bf{\text{Solution Details:}}$ In the equation above, $a= a ,$ $b= 8 ,$ and $c= 1 .$ Using the Discriminant Formula which is given by $b^2-4ac,$ the discriminant is \begin{array}{l}\require{cancel} (8)^2-4(a)(1) \\\\= 64-4a .\end{array} Equating the discriminant to $0$ so that the given equation will have $1$ rational solution, then \begin{array}{l}\require{cancel} 64-4a=0 .\end{array} Using the properties of equality to isolate the variable results to \begin{array}{l}\require{cancel} -4a=-64 \\\\ a=\dfrac{-64}{-4} \\\\ a=16 .\end{array} Hence, the given equation has $1$ rational solution when $a=16 .$