Lesson PROOF of quadratic formula by completing the square

This lesson will prove that quadratic equations can be solved by "completing the square", and I will show you how it is done.

Consider this quadratic equation:
                (1)
assuming coefficients a, b and c as real numbers.

Try to use "completing the square":
add and subtract :



Rewrite the equation



Factor for a



Factor the complete square





Multiply both sides by
  (2)
Expression
                      (3)
is the discriminant of the quadratic equation (1).
Assuming the discriminant is positive or zero (no negative),
take the square root for both sides. Note we are still in the area of real numbers.
or
or
          (4)

Conclusion:

1) If the discriminant of the quadratic equation (1) with real coefficients a, b and c is positive,
then the equation has two different real roots given by formula (4).

2) If the discriminant of the quadratic equation (1) is equal to zero,
then the equation has one real root given by formula (4).

3) If the discriminant of the quadratic equation (1) is negative,
then the equation has no real roots.
It follows from formula (2) because the left side as a square of the real number can not be negative.

Note for the area of complex numbers.

Let us consider the quadratic equation (1) in the area of complex numbers.
This means that coefficients a, b and c are complex numbers and we search a solution among complex numbers.
Then the equation always has two roots in the area of complex numbers.
These roots are given by formula (4).
If the discriminant (3) is zero, then two roots are actually equal each to other.