Answer:
2 rational solutions
Explanation:
the degree of the polynomial (the highest exponent of the variable) defines how many solutions there are.
that means for how many values of x the result y is 0 (how many x-axis interceptions there are or should be).
so, a quadratic equation has 2 such solutions.
there are a few different cases how these solutions present themselves :
"normal case" : all the solutions are real (not only rational) numbers, and the curve has that many different x- acid interceptions.
a turn of the curve leads to only "touching" the x-axis : this is just a special case that the curve is "too high" or "too low" and the regular 2 intersects merge into 1. there are formally still 2 solutions, but they are identical.
there can be "S-curves" especially for higher degrees (3 and up) with the center smoothing in with the x-axis : this is similar but can merge even more than 2 solutions into 1.
the curve is really "too high" or "too low" and never even touches the x-axis : then the solutions are irrational.
the general solution for a quadratic equation is
x = (-b ± sqrt(b² - 4ac))/(2a)
the discriminate is (b² - 4ac).
if this is a positive perfect square, then the square root is a perfect integer. in fact, 2 perfect integers (+ and -).
and so we get two solutions with integer/integer and they are therefore 2 rational solutions.