Answer
The given equation has one rational solution (with multiplicity 2).
Work Step by Step
Distribute $x$ to obtain:
$$x(9x)+x(6)=-1
\\9x^2+6x=-1
\\9x^2+6x+1=0$$
This this equation has $a=9$, $b=6$, and $c=1$.
RECALL:
(1) The discriminant is equal to $b^2-4ac$.
(2) A quadratic equation has the following types of solutions based on the value of the discriminant:
(a) when $b^2-4ac\gt0$, the equation has two unequal rational solutions;
(b) when $b^2-4ac=0$, the equation has one, repeated rational solution; and
(c) when $b^2-4ac\lt0$, the equation has two complex number solutions;
The discriminant of the equation above is:
$$b^2-4ac =6^2 - 4(9)(1) = 36-36=0$$
The discriminant is zero.
Thus, the given equation has one rational solution (with multiplicity 2).