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We want to use the discriminant of the given quadratic equation to determine the number of real solutions. In the Quadratic Formula, ${\color{#FF0000}{b^2-4ac}}$ is the discriminant. $\begin{gathered} x=\dfrac{\text{-} b\pm\sqrt{{\color{#FF0000}{b^2-4ac}}}}{2a} \end{gathered}$ If we just want to know the number of real solutions, and not the solutions themselves, we only need to work with the discriminant. Let's first identify the values of ${\color{#0000FF}{a}},$ ${\color{#FF0000}{b}},$ and ${\color{#009600}{c}}.$ $\begin{gathered} {\color{#0000FF}{9}}x^2+{\color{#FF0000}{12}}x+{\color{#009600}{4}}=0 \end{gathered}$ Finally, let's evaluate the discriminant.
$b^2-4ac$
${\color{#FF0000}{12}}^2-4\cdot{\color{#0000FF}{9}}\cdot{\color{#009600}{4}}$
$144-4\cdot9\cdot4$
$144-144$
$0$
Since the discriminant is $0,$ the quadratic equation has one real solution.