Why is the discriminant determines the importances of the number of solutions?

Answer 1

In the quadratic formula, you have -b +- square root of ( b^2 - 4ac) all over 2a. There are three cases:

No values:

If the discriminant is negative, the square root of b^2 -4ac has no values, because we cannot take the square root of a negative number.

One value:

If it is zero, than the square root is also zero. This means that there is one solution because +- 0 is always just 0. (-b+-0)/2a only has one value.

2 values:

if the discriminant is positive then it has a square root. BUT, there is a difference between here and the last step. (-b +- discriminant )/ 2a has 2 values: one for the positive and one for the negative.

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Minor amendment:

-b +- square root of ( b^2 - 4ac) all over 2a is the quadratic formula.

The discriminant is simply b2 - 4ac

If b2 - 4ac > 0 then there are two distinct real roots to the quadratic;

If b2 - 4ac = 0 then there is one real roots to the quadratic (or two identical roots);

If b2 - 4ac < 0 then there are no real roots to the quadratic.

The values of the roots are exactly as described in the earlier answer.