Answer
$2$ real rational distinct solutions
Work Step by Step
Bring the equation to the standard form by moving all terms on one side:
$$\begin{align*}
9x^2+3x-2&=0.
\end{align*}$$
To determine the number and the type of the solutions for the equation $ax^2+bx+c=0$, we have to calculate the discriminant $\Delta=b^2-4ac$ and compare it to zero.
Identify $a$, $b$, $c$:
$$\begin{align*}
a&=9\\
b&=3\\
c&=-2.
\end{align*}$$
Calculate the discriminant:
$$\begin{align*}
\Delta=3^2-4(9)(-2)=81.
\end{align*}$$
Because $\Delta>0$, the equation has $2$ real distinct solutions.
Because $\delta$ is a perfect square, the solutions are rational.
So the equation has $2$ real rational distinct solutions.
