## Intermediate Algebra (12th Edition)

$c=9$
$\bf{\text{Solution Outline:}}$ To find the value of $c$ such that the given equation, $4m^2+12m+c=0 ,$ will have $1$ rational solution, equate the discriminant to $0$ and solve for the variable. $\bf{\text{Solution Details:}}$ In the equation above, $a= 4 ,$ $b= 12 ,$ and $c= c .$ Using the Discriminant Formula which is given by $b^2-4ac,$ the discriminant is \begin{array}{l}\require{cancel} (12)^2-4(4)(c) \\\\= 144-16c .\end{array} Equating the discriminant to $0$ so that the given equation will have $1$ rational solution, then \begin{array}{l}\require{cancel} 144-16c=0 .\end{array} Using the properties of equality to isolate the variable results to \begin{array}{l}\require{cancel} -16c=-144 \\\\ c=\dfrac{-144}{-16} \\\\ c=9 .\end{array} Hence, the given equation has $1$ rational solution when $c=9 .$