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Why does the part of the quadratic formula under the square root determine the roots (f(x)=0) of the function? Also, what does "complex roots" mean? How can you have complex numbers on the plane (where x and y are real numbers)?
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You can "identify" a complex number with a pair of real number, in the sense that there is a way to convert between them and a way to convert operations on complex numbers into matching operations on pair of real numbers, but complex plane is technically different from the real plane. For example, a number 2+i can be identified with pair of real number (2,1).

Complex roots means roots that are complex number.

The part under the square root is the discriminant. The discriminant of a quadratic equation is the square of the difference between 2 roots times the square of the leading coefficient a. So square root of discriminant divided by a is just the difference between 2 roots. If you know the difference and the sum of 2 numbers, you can calculate them easily (the sum of 2 root is -b/a)

In general, for arbitrary polynomials, the discriminant is the square of product of all difference between every pair of root times the leading coefficient raised to power of the degree of the polynomial.
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