Deriving The Quadratic Formula

First, let's briefly discuss solving quadratic equations using a method called:

Completing The Square

Let's see what a typical perfect square looks like.

(X + n)² =   X² + 2nx + n²

Note the rightmost term (n²) is related to the coefficient of x (2n) by:
(This will become apparent when you see Step 3 below)

Now, let's choose an example Quadratic Equation:

4X² + 12X -16 = 0

Solving this by "completing the square" is as follows:

1) Move the "non X" term to the right:

4X² + 12X = 16

2) Divide the equation by the coefficient of X² which in this case is 4
X² + 3X = 4

3) Now here's the "completing the square" stage in which we:

      take the coefficient of X
      divide it by 2
      square that number
      then add it to both sides of the equation.

In our sample problem

      the coefficient of X is 3
      dividing this by 2 equals 1.5
      squaring this number equals (1.5)² = 2.25
      Now, adding that to both sides of the equation, we have:

X² + 3X + 2.25 = 4 + 2.25

4) Finally, we can take the square root of both sides of the equation and we have:

X + 1.5 = Square Root (4 + 2.25)
X = Square Root (6.25) -1.5
X = 2.5 -1.5
X = 1.0
Let's not forget that the other square root of 6.25 is -2.5 and so the other root of the equation is:
(-2.5 -1.5) = -4

The Quadratic Formula

We can follow precisely the same procedure as above to derive the Quadratic Formula.   All Quadratic Equations have the general form:

aX² + bX + c = 0


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