The Quadratic Formula and Discriminant
October 8, 2026

[00:00:05] so first step is we're going to subtract C from both sides. Okay. And now, where are
[00:00:27] Where are we at? Divide by a. We need to turn the leading coefficient into one, so we divide
[00:00:33] both sides by the leading coefficient. Alright, now we've got to complete the square. Okay,
[00:01:04] We take this, which is no longer our b value, but it is like our new b value.
[00:01:11] We take that, divide by two, and square it. That becomes (b over two a) squared. We could write it like this.
[00:01:35] Or we can leave it like this, either one. This is what we add to both sides: b squared over four a squared.
[00:01:47] And I'm going to write on the left side here. You'll see why in a minute. Okay. We normally write it just to the right over here.
[00:02:02] If I put it here, you will see. Now this factors. What does this factor to? x plus b over two a. It is always this thing divided by two.
[00:02:23] So B over 2A. B over 2A. And then the right side, we're going to need to get a common denominator. So the common denominator is going to be 4A squared.
[00:02:45] So this needs a 4A over 4A. That's B squared over 4A squared minus 4AC over 4A squared.
[00:03:15] Okay. Now we have something squared equals some number. Is it just a number? What do we use next? The square root principle. The something that is squared is plus or minus the square root.
[00:03:53] And what? Let's subtract B over 2A. And I put it on the left side.
[00:04:26] Then this denominator simplifies. What does that simplify to?
[00:04:31] Two a. Now we have a common denominator, so we can make one fraction.
[00:05:12] And there it is. Okay. We just derived or proved that all we got to do is know the coefficients A, B, and C.
[00:05:28] And all we do is plug it into this formula. And this will give us our two solutions. This is what's known as the quadratic formula. The very famous formula. It works for every single quadratic equation.
[00:05:46] Okay. So this is called the quadratic formula.
[00:05:54] In fact, we never have to use completing the square again. We did it for all quadratics. Instead of going through the whole thing, we plug the numbers a, b, and c into here and simplify.
[00:06:14] But whenever you want to solve by factoring, you do not have to do this. If the quadratic can be solved by factoring, we usually do that if it is quick. If it is not quick and you cannot figure it out, we go to the quadratic formula.
[00:06:35] Okay. Before we use the quadratic formula, what do we have to do? Put it in the standard form. It's called Ax squared plus Bx plus C equals zero. So we have to subtract 3x from both sides.
[00:07:25] Okay.
[00:07:26] x equals negative b plus or minus the square root of b squared minus four a c, all over two a.
[00:07:55] We have negative b, the negative of negative three, plus or minus the square root of b squared: negative three squared minus four times a, which is one, times c, which is negative four, all over two times one.
[00:08:27] And now we simplify.
[00:08:43] Four times negative four is negative sixteen, so subtracting negative sixteen gives plus sixteen, all over two.
[00:09:09] And so again, this is two separate equations. Well, I'm going to write one more first. 3 plus or minus 5 over 2, which is x is 3 plus 5 over 2, or x is 3 minus 5 over 2.
[00:09:59] Okay.
[00:10:01] Our solutions are integers. That tells us it could have been solved by factoring. If the thing inside the square root is not a perfect square, it could be solved by the quadratic formula or completing the square.
[00:10:23] This makes sense. Okay. That's the first one using a formula. We got to memorize the formula. This is very important.
[00:10:35] One way to memorize it is by this song.
[00:10:40] x equals negative b plus or minus the square root of b squared minus four a c, all over two a.
[00:10:52] Okay, now with your pen in hand on your paper, we're going to say.
[00:11:05] We are going to sing and write the formula as we go.
[00:11:15] Here we go.
[00:11:19] x equals negative b plus or minus the square root of b squared minus four a c, all over two a.
[00:11:33] This is already in standard form. What is my a?
[00:11:38] Two. b?
[00:11:41] Negative five. c?
[00:11:45] So first let's write it with the letters in there and then we'll write with the numbers plugged in. So here we go.
[00:12:33] Okay.
[00:13:22] Can we simplify the square root of thirty-three?
[00:13:30] And we're done. That's it. These are this is my solution set.
[00:13:39] If we ever get this, we're going to learn this later.
[00:13:44] What was my original equation? Two x squared.
[00:13:54] It tells us this does factor.
[00:14:00] The two zeros.
[00:14:03] We subtract these numbers: five minus square root thirty-three, over four.
[00:14:17] Five plus square root thirty-three, over four.
[00:14:25] Except we're missing the leading coefficient. So we have the two out here.
[00:14:30] So that's what this thing factors to two times X minus one of the solutions times X minus the other solution.
[00:14:39] It should make sense, right? You plug in a solution right here. That same number minus that number zero.
[00:14:47] And the other number plus.
[00:14:51] Five plus square root thirty-three, over four, minus that number.
[00:14:56] And that's it. So every polynomial does factor to exactly the leading coefficient, the A, times X minus one of the solutions times X minus the other solutions.
[00:15:16] OK, let's see some other. Let's try some more.
[00:15:20] This quadratic has a common factor in all three terms.
[00:15:25] What is the common factor? Two. But I want to show you what happens if we do this.
[00:15:35] We could divide both sides by two and make all the numbers smaller and it would be easier.
[00:15:41] Let's see what happens when we use quadratic formula when there is a common factor in here.
[00:15:50] Negative b. What is b?
[00:15:55] Negative four plus or minus the square root of b squared minus four a c, all over two a.
[00:16:10] Hopefully we go right to that. You do not even have to list a, b, and c: negative b plus or minus the square root of b
[00:16:19] Square minus four times A C.
[00:16:25] And now let's simplify.
[00:16:36] What is that plus sixteen?
[00:17:16] OK, this is the two solutions. However, we can simplify this. What is the square root of thirty two simplified to?
[00:17:26] Four square root two.
[00:17:34] Now both of these have a common factor of four.
[00:17:41] So we can either divide both of these by four.
[00:17:47] It would be negative four over four, plus or minus four square root two over four.
[00:17:56] We get negative one plus or minus square root two.
[00:18:05] The other option is you factor out a four.
[00:18:15] And those simplify.
[00:18:18] Negative one plus or minus square root two.
[00:18:22] So if there is a common factor, you can still leave it in there, except you're definitely going to have to simplify.
[00:18:31] And there will be a common thing that factors out.
[00:18:36] OK, let's try another one.
[00:19:07] x equals negative b plus or minus the square root of b squared minus four a c, all over two a.
[00:19:21] What is B?
[00:19:23] Negative ten.
[00:19:26] Negative ten.
[00:19:27] Negative ten squared.
[00:20:19] Negative ten.
[00:20:28] So our two solutions are ten plus zero over two and ten minus zero over two.
[00:20:37] OK, so we really only have one solution.
[00:20:43] Whenever we end up with just one solution or in other words, when this thing is zero, it's the only time you get one solution.
[00:20:52] It does tell us that this is a perfect squared trinomial.
[00:20:59] OK, this factors to X minus five times X minus five, which is X minus five squared.
[00:21:10] Whenever the thing inside the radical of the quadratic formula is zero,
[00:21:20] We're only going to get one solution.
[00:21:23] And it tells us the quadratic is a perfect square trinomial.
[00:21:32] Now let's see what happens.
[00:21:56] OK.
[00:22:28] Sometimes b squared minus four a c is a negative number.
[00:22:37] So what does that tell us?
[00:22:40] This is not a real number.
[00:22:47] OK, so there are no real solutions.
[00:22:57] Does it have a solution set? Yes. Yes, and it is the empty set.
[00:23:03] OK, so as soon as B squared minus four AC is a negative, there's no real solution.
[00:23:10] Later on, we will learn something very interesting, and we will not be done here.
[00:23:17] But right now we will be done.
[00:23:21] OK, so let's look one more time at this thing.
[00:23:25] OK, this radicand of the quadratic formula is called the discriminant.
[00:23:36] OK, so this number classifies our solutions.
[00:23:43] OK, if B squared minus four AC is greater than zero, what does that tell us?
[00:23:56] There's two real solutions.
[00:24:10] If the discriminant B squared minus four AC is equal to zero, what does that tell us?
[00:24:20] There's one real solution.
[00:24:25] It also tells us it can be factored: the quadratic is a perfect square trinomial.
[00:24:51] And if B squared minus four AC is less than zero, what does that tell us?
[00:25:02] There are no real solutions.
[00:25:06] OK, and then we tell this even more.
[00:25:14] If B squared minus four AC is a perfect square.
[00:25:27] What does that tell us?
[00:25:30] What's that?
[00:25:31] The solutions are integers?
[00:25:33] Not necessarily because we have to divide at least by two.
[00:25:40] It might not be an integer, but the square root of the perfect square.
[00:25:48] The square root will be an integer.
[00:25:51] Well, it could be.
[00:25:53] Yeah, but it tells us we're going to have two.
[00:25:57] So the square root will simplify since it's a perfect square.
[00:26:00] So we have two rational solutions.
[00:26:10] What's a rational number?
[00:26:15] Any number that can be written as the ratio of two integers.
[00:26:21] OK, if B squared minus four AC is not a perfect square.
[00:26:38] Then what?
[00:26:41] Then there are two irrational solutions.
[00:26:54] And your answer is going to have a square root.
[00:26:58] If we want to approximate the solution, we need a calculator to type in the square root, then add negative b and divide by two a.
[00:27:08] OK, so the discriminant tells a lot.
[00:27:12] If it's positive, two real.
[00:27:13] If it's zero, there's one real.
[00:27:16] And the quadratic is a perfect square trinomial.
[00:27:19] If the discriminant is negative, there's no real solutions.
[00:27:23] If it's a perfect square, there's two rational.
[00:27:32] And if it's not a perfect square, there's two irrational.
[00:27:35] If b squared minus four a c is a perfect square, it tells us we could have solved by factoring.