Worked example: Rewriting and solving equations by completing the square (IMRA 2026 publisher-initiated change)

0:00- [Instructor] So let's see if we can solve this quadratic equation right over here, x squared minus 2x minus 8 is equal to 0. And actually, they're cutting down some trees outside so my apologies if you hear some chopping of trees. Well, I'll try to ignore it myself.

0:15All right, so back to the problem at hand. And there's actually several ways that you could attack this problem. You could just try to factor the left-hand side and go that way. But the way we're going to tackle it is by completing the square. And what does that mean?

0:30Well, that means that I wanna write, I wanna write the left-hand side of this equation into the form x plus a squared plus b. And as we'll see if we can write the left hand in this form, then we can actually solve it in a pretty straightforward way. So let's see if we could do that.

0:50Let's just remind ourselves how we need to rearrange the left-hand side in order to get it to this form. If I were to expand out x plus a squared, let me do that in a different color. So if I were to expand out x plus a squared, that is x squared plus 2ax, make sure that plus sign you can see, plus 2ax plus a squared.

1:11And of course, you still have that plus b there, plus b. So let's see if we can write this in that form. So what I'm going to do, and this is what you typically do when you try to complete the square, I'll write the x squared minus 2x. Now, I'm gonna have a little bit of a gap and I'm going to have minus 8 and I have another little bit of a gap and I'm gonna say equals 0.

1:33So I just rewrote this equation, but I gave myself some space so I can add or subtract some things that might make it a little bit easier to get into this form. So if we just match our terms, x squared, x squared, 2ax, -2x. So if this is 2ax, that means that 2a is -2, 2a is equal to -2, or a is equal to -1.

2:01Another way to think about it, your a is going to be half of your first degree coefficient, so or the coefficient on the x term. So the coefficient of the x term is a -2, half of that is a -1. And then we wanna have, and then we want to have an a squared. So if a is -1, a squared would be plus 1 so let's throw a plus 1 there.

2:25But like we've done said before, we can't just willy-nilly add something on one side of the equation without adding it to the other or without subtracting it again on that same side. Otherwise, you're fundamentally changing, you're fundamentally changing the truth of the equation.

2:39So if I add 1 on that side, I either have to add 1 on the... If I add 1 on the left side, I either have to add 1 on the right side to make the equation still hold true, or I could add 1 and subtract 1 from the left-hand side, so I'm not really changing the value of the left-hand side.

2:53All I've done is added 1 and subtracted 1 from the left-hand side. Now, why did I do this again? Well, now I've been able, I haven't changed its value, I just added and subtracted the same thing, but this part of the left-hand side now matches this pattern right over here.

3:10It's x squared plus 2ax where a is -1, so it's minus 2x plus a squared plus -1 squared. And then this, this part right over here is the plus b, so we already know that b is equal to -9.

3:28- 8 minus 1 is -9, and so that's going to be our b right over there. And so we can rewrite this as what I squared off in green, that's gonna be x plus a squared. So we could write it as x plus and I could write a is -1. Actually like, let me, I could write it like that first, x plus a squared or x plus -1.

3:54Well, that's just x minus 1 so I'm just gonna write it as x minus 1 squared. And then we have minus 9. Minus 9 is equal to 0, is equal to 0. And then I can add 9 to both sides, so I just have this squared expression on the left-hand side. So let's do that, let me add 9 to both sides, and what I am going to be left with.

4:25So let me just, on the left-hand side, those cancel out, that's why I added the 9. I'm just gonna be left with the x minus 1 squared, it's going to be equal to, it's going to be equal to on this side, 0 plus 9 is 9. So if x minus 1, let me do that blue color, so it's gonna be 9.

4:50And so if x minus 1 squared is 9, if I have something squared is equal to 9, that means that that something is either going to be the positive or the negative square root of 9, so it's either going to be + or -3. So we can say x minus 1 is equal to +3 or x minus 1 is equal to -3.

5:10And you could see that here. If x minus 1 is 3, 3 squared is 9. If x minus 1 is -3,

5:15- 3 squared is 9. And so here we can just add 1 to both sides of this equation. Add 1 to both sides of this equation, and you get x is equal to 4. Or x is if we add 1 to both sides of this equation, we get.. My digital ink is acting up, I don't know. All right, then we get x is equal to -3 plus 1 is -2.

5:44So x could be equal to 4 or x could be equal to -2, and we're done. Now, some of you might be saying, "Well, why did we go through the trouble of completing the square? I might have been able to just factor this and then solve it that way." And you could have actually for this particular problem, but completing the square is very powerful because you can actually always apply this.

6:01And what you will, in the future, you will learn the quadratic formula, and the quadratic formula actually comes directly out of completing the square. In fact, when you're applying the quadratic formula, you're essentially applying the result of completing the square.

6:15So hopefully you found that fun.