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Sal solves challenging quadratic equations like (4x+1)²-8=0 by taking the square root of both sides. Created by Sal Khan and CK-12 Foundation.
Video transcript
In this video, I'm going to do several examples of quadratic equations that are really of a special form, and it's really a bit of warm-up for the next video that we're going to do on completing the square. So let me show you what I'm talking about. So let's say I have 4x plus 1 squared, minus 8 is equal to 0. Now, based on everything we've done so far, you might be tempted to multiply this out, then subtract 8 from the constant you get out here, and then try to factor it. And then you're going to have x minus something, times x minus something else is equal to 0. And you're going to say, oh, one of these must be equal to 0, so x could be that or that. We're not going to do that this time, because you might see something interesting here. We can solve this without factoring it. And how do we do that? Well, what happens if we add 8 to both sides of this equation? Then the left-hand side of the equation becomes 4x plus 1 squared, and these 8's cancel out. The right-hand becomes just a positive 8. Now, what can we do to both sides of this equation? And this is just kind of straight, vanilla equation-solving. This isn't any kind of fancy factoring. We can take the square root of both sides of this equation. We could take the square root. So 4x plus 1-- I'm just taking the square root of both sides. You take the square root of both sides, and, of course, you want to take the positive and the negative square root, because 4x plus 1 could be the positive square root of 8, or it could be the negative square root of 8. So 4x plus 1 is equal to the positive or negative square root of 8. Instead of 8, let me write 8 as 4 times 2. We all know that's what 8 is, and obviously the square root of 4x plus 1 squared is 4x plus 1. So we get 4x plus 1 is equal to-- we can factor out the 4, or the square root of 4, which is 2-- is equal to the plus or minus times 2 times the square root of 2, right? Square root of 4 times square root of 2 is the same thing as square root of 4 times the square root of 2, plus or minus the square root of 4 is that 2 right there. Now, it might look like a really bizarro equation, with this plus or minus 2 times the square of 2, but it really isn't. These are actually two numbers here, and we're actually simultaneously solving two equations. We could write this as 4x plus 1 is equal to the positive 2, square root of 2, or 4x plus 1 is equal to negative 2 times the square root of 2. This one statement is equivalent to this right here, because we have this plus or minus here, this or statement. Let me solve all of these simultaneously. So if I subtract 1 from both sides of this equation, what do I have? On the left-hand side, I'm just left with 4x. On the right-hand side, I have-- you can't really mathematically, I mean, you could do them if you had a calculator, but I'll just leave it as negative 1 plus or minus the square root, or 2 times the square root of 2. That's what 4x is equal to. If we did it here, as two separate equations, same idea. Subtract 1 from both sides of this equation, you get 4x is equal to negative 1 plus 2, times the square root of 2. This equation, subtract 1 from both sides. 4x is equal to negative 1 minus 2, times the square root of 2. This statement right here is completely equivalent to these two statements. Now, last step, we just have to divide both sides by 4, so you divide both sides by 4, and you get x is equal to negative 1 plus or minus 2, times the square root of 2, over 4. Now, this statement is completely equivalent to dividing each of these by 4, and you get x is equal to negative 1 plus 2, times the square root 2, over 4. This is one solution. And then the other solution is x is equal to negative 1 minus 2 roots of 2, all of that over 4. That statement and these two statements are equivalent. And if you want, I encourage you to-- let's substitute one of these back in, just so you feel confident that something as bizarro as one of these expressions can be a solution to a nice, vanilla-looking equation like this. So let's substitute it back in. 4 times x, or 4 times negative 1, plus 2 root 2, over 4, plus 1 squared, minus 8 is equal to 0. Now, these 4's cancel out, so you're left with negative 1 plus 2 roots 2, plus 1, squared, minus 8 is equal to 0. This negative 1 and this positive 1 cancel out, so you're left with 2 roots of 2 squared, minus 8 is equal to 0. And then what are you going to have here? So when you square this, you get 4 times 2, minus 8 is equal to 0, which is true. 8 minus 8 is equal to 0. And if you try this one out, you're going to get the exact same answer. Let's do another one like this. And remember, these are special forms where we have squares of binomials in our expression. And we're going to see that the entire quadratic formula is actually derived from a notion like this, because you can actually turn any, you can turn any, quadratic equation into a perfect square equalling something else. We'll see that two videos from now. But let's get a little warmed up just seeing this type of thing. So let's say you have x squared minus 10x, plus 25 is equal to 9. Now, once again your temptation-- and it's not a bad temptation-- would be to subtract 9 from both sides, so you get a 0 on the right-hand side, but before you do that, just inspect this really fast. And say, hey, is this just maybe a perfect square of a binomial? And you see-- well, what two numbers when I multiply them I get positive 25, and when I add them I get negative 10? And hopefully negative 5 jumps out at you. So this expression right here is x minus 5, times x minus 5. So this left-hand side can be written as x minus 5 squared, and the right-hand side is still 9. And I want to really emphasize. I don't want this to ruin all of the training you've gotten on factoring so far. We can only do this when this is a perfect square. If you got, like, x minus 3, times x plus 4, and that would be equal to 9, that would be a dead end. You wouldn't be able to really do anything constructive with that. Only because this is a perfect square, can we now say x minus 5 squared is equal to 9, and now we can take the square root of both sides. So we could say that x minus 5 is equal to plus or minus 3. Add 5 to both sides of this equation, you get x is equal to 5 plus or minus 3, or x is equal to-- what's 5 plus 3? Well, x could be 8 or x could be equal to 5 minus 3, or x is equal to 2. Now, we could have done this equation, this quadratic equation, the traditional way, the way you were tempted to do it. What happens if you subtract 9 from both sides of this equation? You'll get x squared minus 10x. And what's 25 minus 9? 25 minus 9 is 16, and that would be equal to 0. And here, this would be just a traditional factoring problem, the type that we've seen in the last few videos. What two numbers, when you take their product, you get positive 16, and when you sum them you get negative 10? And maybe negative 8 and negative 2 jump into your brain. So we get x minus 8, times x minus 2 is equal to 0. And so x could be equal to 8 or x could be equal to 2. That's the fun thing about algebra: you can do things in two completely different ways, but as long as you do them in algebraically-valid ways, you're not going to get different answers. And on some level, if you recognize this, this is easier because you didn't have to do that little game in your head, in terms of, oh, what two numbers, when you multiply them you get 16, and when you add them you get negative 10? Here, you just said, OK, this is x minus 5-- oh, I guess you did have to do it. You had to say, oh, 5 times 5 is 25, and negative 10 is negative 5 plus negative 5. So I take that back, you still have to do that little game in your head. So let's do another one. Let's do one more of these, just to really get ourselves nice and warmed up here. So, let's say we have x squared plus 18x, plus 81 is equal to 1. So once again, we can do it in two ways. We could subtract 1 from both sides, or we could recognize that this is x plus 9, times x plus 9. This right here, 9 times 9 is 81, 9 plus 9 is 18. So we can write our equation as x plus 9 squared is equal to 1. Take the square root of both sides, you get x plus 9 is equal to plus or minus the square root of 1, which is just 1. So x is equal to-- subtract 9 from both sides-- negative 9 plus or minus 1. And that means that x could be equal to-- negative 9 plus 1 is negative 8, or x could be equal to-- negative 9 minus 1, which is negative 10. And once again, you could have done this the traditional way. You could have subtracted 1 from both sides and you would have gotten x squared plus 18x, plus 80 is equal to 0. And you'd say, hey, gee, 8 times 10 is 80, 8 plus 10 is 18, so you get x plus 8, times x plus 10 is equal to 0. And then you'd get x could be equal to negative 8, or x could be equal to negative 10. That was good warm up. Now, I think we're ready to tackle completing the square.