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If a quadratic function is 0 for any value of the variable, then that value is a solution.
If you have a quadratic function with real coefficients then it can have: two distinct real roots, or a real double root (two coincidental roots), or no real roots. In the last case, it has two complex roots which are conjugates of one another.
The real solutions are the points at which the graph of the function crosses the x-axis. If the graph never crosses the x-axis, then the solutions are imaginary.
Yes, but in this case, the coefficients of the polynomial can not all be real.
The domain and range can be the whole of the real numbers, or some subsets of these sets.