The give equation has two unequal rational number solutions.
Work Step by Step
Add $3$ to both sides to obtain: $$-6x^2+2x+3=0$$ This this equation has $a=-6$, $b=2$, and $x=3$. 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: $$b^2-4ac = 2^2 - 4(-6)(3) = 4+72=76$$ The discriminant is positive. Thus, the give equation has two unequal rational number solutions.