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An equation can be written given the solutions of the equation in the following way. Assume an equation f(x) = 0 has the roots x1 and x2.
The corresponding equation would be (x - x1)(x - x2) = 0
=> x^2 - (x1 + x2)*x + x1*x2 = 0
For a higher order equation with solutions x1, x2, x3..., the corresponding equation is (x - x1)(x - x2)(x - x3)... = 0. This can be simplified to give an equation in the standard form.
The same is applicable for the creation of an equation of any order.
I don't know what might be meant by "rational equation." but x1 and x2 might also be the solutions to
|x -a| = b, or
(x - a)^2 + (y - b)^2 = k
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