The give equation has two complex number solutions.
Work Step by Step
Add $2x$ and $6$ to both sides to obtain: $$8x^2+2x+6=-2x-6+2x+6 \\8x^2+2x+6=0$$ This this equation has $a=8$, $b=2$, and $x=6$. 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: $$b^2-4ac = 2^2 - 4(8)(6) = 4-192=-188$$ The discriminant is negative. Thus, the give equation has two complex number solutions.