How to Derive the Quadratic Formula

First, we need to review a technique called "completing the square". Suppose we have the following equation: 

2x^2 + 12x = 14 

To make this a perfect square, we need to get it into the form of a^2 + 2ab + b^2. We can do that by first dividing by 2.

x^2 + 6x = 7

Next, we divide the coefficient of the middle term (which is 6) by 2, then we square it and add it to both sides.

x^2 + 6x + 9 = 16

We can now rewrite the left side as a binomial.

(x + 3)^2 = 16

If we solve, we get x = 1 and x = -7. 

We can apply the same process to the general form of a quadratic equation. 

ax^2 + bx + c = 0

First, we divide by a.

x^2 + (b/a)x + c/a = 0

Then we move c/a to the right side of the equation. 

x^2 + (b/a)x = -(c/a)

We complete the square by dividing the coefficient of the middle term (b/a) by 2, square it, and add it to both sides.

x^2 + (b/a)x + b^2/(4a^2) = b^2/(4a^2) - (c/a)

The left side can now be rewritten.

(x + b/(2a))^2 = b^2/(4a^2) - (c/a)

To rewrite the right side as a fraction, we multiply the top and bottom of c/a by 4a.

(x + b/(2a))^2 = (b^2 - 4ac)/(4a^2)

Next we take the square root of both sides.

x + b/(2a) =  +/- (sqrt(b^2 - 4ac))/2a

Move the b/(2a) to the right.

x = b/(2a) +/- (sqrt(b^2 - 4ac))/2a

Since both terms on the right are divided by 2a, we can combine them into a single fraction.

x = (-b +/- (sqrt(b^2 - 4ac))/2a)

We now have the famous quadratic formula, albeit in somewhat different notation. It's good to know where formulas come from and deriving them on your own is excellent math practice. The whole process of math is simply building a chain of if-then statements. If this is true, then this true, and so this is true, and so on. Add enough links to that chain of reasoning and it will stretch out long enough to reach some very interesting truths.