The given equation has two unequal rational number solutions.
Work Step by Step
Add $26x$ to both sides to obtain: $$16x^2+3+26x=-26x+26x \\16x^2+26x+3=0$$ This this equation has $a=16$, $b=26$, and $c=3$. 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: $$b^2-4ac =26^2 - 4(16)(3) = 676-192=484$$ The discriminant is positive. Thus, the given equation has two unequal rational number solutions.