How to Derive the Quadratic Formula

In this video, I want to look at a shortcut for solving certain types of polynomial equations. Now, the polynomial types that we’ve mostly looked at so far have been ones where the highest power of x is 2. We have something like a times x squared plus b times x plus c equals 0. This type of polynomial equation is also known as a quadratic equation.

A quadratic equation like this is of a predictable form. We just have two different powers of x here and we have 3 numerical constants in addition to 0. What we’re looking for here is a shortcut to where we can solve an equation that looks like this without having to go through completing this square every time.

What we’re going to do is we’re going to derive an expression for x in terms of these three variables, a, b, and c that should work to solve any quadratic equation of this form. The first thing we want to do is get x squared down to itself without an “a” in front of it. We’re going to divide each term by a. That gives us x squared plus b over a times x plus c over a equals 0.

Then, we want to move this c term to the other side of the equation. That will give us x squared plus b over a x equals negative c over a. The next step is to figure out what our number that will be added to this equation here is going to be. Whatever this numerical constant is here, we’ll call it “n” in this case.

We need to figure out what that’s going to be. The formula for figuring out what n is going to be is going to be whatever this coefficient of x is divided by 2 squared. That gives us n equals b squared over 4a squared, because we divided it by 2, b over 2a is this divided by two, and then squared is b squared over 4a squared.

What we’re going to add to both sides here is b squared over 4a squared. That gives us x squared plus b over a times x plus b squared over 4a squared equals (will go ahead and, well let’s just write it as we have it for now), so b squared over 4a squared minus c over a.

Now, this side of the equation we can rewrite, because we picked our constant here such that this side of the equation would equal x plus b over 2a squared. In this side of the equation, we can combine the two fractions if we multiply this fraction top and bottom by 4a so that they have a common denominator of 4a squared. We’ll go ahead and multiply by 4a over 4a.

That gives us b squared minus 4a times c divided by 4a squared. Now we have just a single term on both sides of the equation. We can go ahead and take the square root of both sides, and that will give us, on this side, it will give us x plus b over 2a. On this side, it will give us- we can’t take the square root of the top directly, but we can take the square to the bottom.

4a squared, or the square root of 4a squared is 2a. We have square root of b squared minus 4a c over 2a here. Like we have to do because we took the square root of both sides and we didn’t know whether this is going to be the positive or negative square root, we have to add a plus or minus in front of one of these terms.

We’ll put it over here. To finish solving for x, we have to subtract b over 2a from both sides. That gives us x equals negative b over 2a plus or minus the square root of B squared minus 4a c over 2a. We can combine this. This is negative b plus or minus the square root of b squared minus 4a c over 2a, because they have a common denominator here in 2a.

This right here, negative b plus or minus the square root of b squared minus 4a c divided by 2a, this is a formula that we can use to solve for x in any quadratic equation. It’s going to give us, in general, two solutions because we have a plus and a minus, and both of these solutions should be correct. This is the derivation of what’s known as the quadratic formula. And we can use this to solve for x directly.