To derive the quadratic formula you take the standard form for a quadratic equation:
and then solve for x by "completing the square" and using square roots. I hope you have learned about completing the square because it is too complicated to explain here. The normal procedure is to:
Move the constant term to the other side of the equation (to get it "out of the way").
If the leading coefficient, "a", is not a 1 then either factor it out or divide both sides by it.
Find half of the coefficient of x.
Add the square of the half (from the previous step) to each side.
Rewrite the side with the as a binomial square (in the form: (x + "the half")^2)
Let's see this in action:
1. Get the constant term "out of the way" (by subtracting "c"):
2. With a quadratic equation (only one squared term) we can divide both sides by "a":
which simplifies:
3. Find half of the coefficient of x. Half of b/a would be:
4. Add the square of the half to both sides. Squaring gives us . Adding this to each side:
5. Rewrite as a binomial square.
Now that we have completed the square we can solve for x. We start with the square root of each side (remembering the positive and negative square roots by using the "plus or minus" sign): = +
Adding we get:
We have solved for x. But our equation does not look like the normal quadratic formula. So we will use some algebra to simplify and rearrange the right side so it looks right. First we will get common denominators in the square root so we can combine the terms:
Combining:
Next we can use a property of radicals to make the square root of the fraction a fraction of square roots:
The square root in the denominator is a perfect square:
The two remaining fractions have the same denominator so we can combine them:
which is the "normal" quadratic formula.
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