Hey everyone. You just finished learning 3 different methods of solving quadratic equations, and you might be wondering how there's anything left and why we have yet another method to learn. But the quadratic formula that we're going to talk about now is really great because it's going to work for any quadratic equation. So even if you were to forget every single other method, as long as you remember the quadratic formula, you're going to be able to solve any quadratic equation that gets thrown at you. So let's go ahead and jump in.

The quadratic formula is based on the standard form of a quadratic equation ax2+bx+c, and we're going to use a, b, and c in order to compute our solutions. So the quadratic formula is −b±b2−4ac2a. Now that formula might look a little bit complicated right now and, unfortunately, it is something that you're going to have to memorize. But a way that I was able to memorize the quadratic formula was by using the quadratic formula song. Now, I'm not going to sing it for you right now, but it's going to be something that you should search up on your own to commit this formula to memory.

So when do we want to use the quadratic formula? Well, like I said, you can use the quadratic formula whenever you want, and some clues that you might want to use it are that you can't easily factor or you're just otherwise unsure what method to use. Let's go ahead and take a look at an example here. So I have x2+2x−3 is equal to 0. Let's go ahead and take a look at our first step, which is to write our equation in standard form. Now it looks like my equation is already in standard form, all of my terms are on the same side in the in descending order of power, so I can go ahead and move on to step 2.

Now step 2 is simply to just plug everything into my quadratic formula, so let's go ahead and do that here. I'm first going to label a, b, and c in my equation so that it's easy for me to just take them and plug them in. So in this case, I have an invisible one in front of that x squared and that is my a, b is this positive 2, and then c is negative 3. Make sure that you're paying attention to the signs here. So plugging this in, I get −2±162. Now that we've plugged everything in, we've completed step 2, and all that we have left to do from here is algebra.

We're just going to compute and simplify our solutions now. So let's start with what's in our radical here and just simplify that. Now I know that the square root of 16 is just 4, so this becomes −2±42. Now is when we're going to want to split our solution into the plus and the minus. So let's go ahead and do that. This splits into −2+42 and −2−42. And now I can just simplify those solutions down. So −2 + 4 is going to give me positive 2 over 2, which is just equal to 1. And then −2 − 4 is going to give me −6 divided by 2, which is going to give me −3. And I'm done. These are my solutions. So for this quadratic equation, my solutions are x equals 1 and x equals −3 and I'm completely done there.

I know that the quadratic formula can look a little intimidating at first but it really just comes down to the algebra once you plug everything in. Let's go ahead and take a look at one more example. So here I have x2−5x is equal to −1. Let's go ahead and start back at step 1, which is to write our equation in standard form. Now here, it looks like I need to go ahead and move my −1 over. So I can do that by simply adding 1 to both sides. And so this will become x2−5x+1 is equal to 0, because that one canceled on that right side. So step 1 is good. That's in standard form now. We can move on to step 2 and just plug everything in. So again, I'm going to go ahead and label a, b, and c here. I again have an invisible one for that a, b is negative 5, and c is positive 1.

So let's go ahead and complete step 2 and plug everything in. So I have −−5±212. Okay. So step 2 is done. Now we're just left to simplify. Just do all of that algebra. So let's first start with everything under that radical. Now I can't simplify this anymore. Even if I split it, nothing would simplify because I just have that square root of 21 can't go anywhere from there. So my solutions are just 5 plus or minus the square root of 21 over 2. Now I can of course split these into their plus and minus just to show both answers separately and I would just write 5 + 21 over 2 and 5 − 21 over 2. Now, as you saw with some of our other methods, you're going to have different types of answers here. So you might have whole numbers, you might also have some combination of fractions and radicals, and either is fine. That's all there is to the quadratic formula let me know if you have any questions.