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).