Strategy in solving quadratic equations

- [Instructor] In this video, we're gonna talk about a few of the pitfalls that someone might encounter while they're trying to solve a quadratic equation like this. Why is it a quadratic equation? Well, it's a quadratic because it has this second degree term right over here and it's an equation because I have something on both sides of an equal sign. So one strategy that people might try is, well, I have something squared, why don't I just try to take the square root of both sides? And if you did that, you would get the square root of x squared plus four x plus three is equal to the square root of negative one. And you immediately see a few problems. Even if this wasn't a negative one here, that's the most obvious problem. But even if this was a positive value here, how do you simplify or how do you somehow isolate the x over here? You've pretty quickly hit a dead end. So just willy nilly, taking the square root of both sides of a quadratic is not going to be too helpful. So I'll put a big X over there. Another strategy that sometimes people will try to go for is to isolate the x squared first. So you could imagine, let me just rewrite it. X squared plus four x plus three is equal to negative one. They might say, let's isolate that x squared by subtracting four x from both sides and subtracting three from both sides. And then what happens? On the left hand side, you do indeed isolate the x squared, and on the right hand side, you get negative four x minus four. And now, someone might say, if I take the square root of both sides, I could get, I'll just write that down. Square root of x squared is equal to, and you could try to take the plus of minus of one side to make sure you're hitting the negative roots. Negative four x minus four. And you could get something like this, you would get x is equal to plus or minus the square root of negative four x minus four, but this still doesn't help you. You still don't know what x is, and it's really not clear what to do with this algebraically. So this is yet another dead end. Now, there's some cases in which this strategy would have worked. In fact, it would have worked if you did not have this first degree term. If you did not have this x term, so to speak. Then this strategy would have worked assuming that there are some solutions. But if you have an x term like this and it doesn't cancel out somehow, you know, if there was another four x on the other side, then you could subtract four x from both sides, and they would disappear. But if can't make these things disappear, this strategy that I've just outlined is not going to be a productive one. Now another strategy that you'll sometimes see people use, especially when they see something like this, let me rewrite it. X squared plus four x plus three is equal to negative one. They immediately go into factoring mode. They say, hey, wait, I think I might be able to factor this. I can think of two numbers that add up to four and whose product is three. Maybe three and one. And then they immediately factor this left hand expression, say that's going to be x plus three times x plus one, and then that's going to be equal to negative one. And then they either are about to make a mistake, this is actually algebraically valid. But they either make a mistake or they realize that they're at a dead end. Because just saying that something times something is equal to negative one doesn't help you a ton. Because it's not clear yet, how you'd solve for x. Another thing, try to do is, is they'll immediately say, okay, therefore x is equal to negative three or x is equal to negative one because negative three will make this first term zero and negative one, or negative one would make the second term zero. But remember, this would only be true if you're multiplying two things and you got zero as their product, then the solutions would be anything that made either one of those terms equal to zero. But that's not what we're dealing with here. Here we're taking the product is equal to negative one. So in order to factor like this and make headway in most cases, you're going to wanna have a zero on the right hand side over here. And that's also true if you're trying to apply the quadratic formula. A lot of folks would say, okay, I see a quadratic equation right over here. Let me just apply the quadratic formula. They say, if I have something of the form ax squared plus bx plus c is equal to zero. The quadratic formula would say that the roots are gonna be x is equal to negative b plus or minus the square root of b squared minus four ac, all of that over two a. And so they'll immediately say, all right, I can recognize a here, as just being a one, there's a one coefficient implicitly there, b is four, c is three. And they'll say, okay, x is equal to negative four plus or minus the square root of b squared, which is 16, minus four times one, times three, all of that over two times one. But there's a problem. The quadratic formula applies when the left hand side is equal to zero. That's not what we have over here. So we're falling into that same pitfall. So everything I just did, none of this is a good idea. So the way to approach this, if you want zero over here, you wanna add one on the right hand side, and if you wanna maintain the equality, you have to add a one on the left hand side. And so you're going to get x squared plus four x plus four is equal to zero. And now you could use the quadratic formula or you could factor. You might recognize two plus two is equal to four. Two times two is equal to four. So you could say x plus two times x plus two is equal to zero. And so in this case, you say, all right, x could be equal to negative two or x could be equal to negative two. (chuckles) So this one only has one solution, x is equal to negative two. But the key is to recognize that you need the zero on the right hand side there, if you wanna use a quadratic formula or if you wanna use factoring and the zero product property.