Hi, I'm Peter Edwards from Bluetutors. We teach children of all ages right from primary school to higher levels, and we find the highest quality tutors. And today, I'm going to teach you some math. We are now going to look at how to solve quadratic equations. We are going to look at two main ways. We will also look at how sometimes you can't solve quadratic equations and just go through a few examples. To start with, I've written one of the most simple equations you can get, and we're going to solve this by factorizing. Now, when someone says that you should factorize a quadratic equation, what they mean is they want you to put in two brackets; an 'x' there, an 'x' there; and a number here, and a number here, such that, when you multiply these two brackets out, it is the same as the expression. So, how do we choose which number goes here and here? Well we want these two numbers to multiply together to give us positive five, and these two numbers to add together to give us negative six. Now, the two numbers which do that are one and five; and, you'll notice, because this has to be negative, they have to add together, they both have to be negative. So, minus one plus minus five gives us negative six. And check, when you multiply minus five with minus one, you'll get positive five. So, once we're in this position, why is that helpful? Well, what we now have is one bracket multiplied by another equals zero, and in that case there are only two possibilities. Either this bracket here is equal to zero, or this bracket here is equal to zero. If this bracket is equal to zero, we know that x minus one equals zero, and therefore x equals one. If this bracket here is equal to zero, we know that x minus five is equal to zero, and therefore x equals five. So, the result of this equation would be that x is equal to one or five. Okay, so, that's the way of solving these equations if you can factorize them, and there are many more difficult factorizations than the one we've just shown. However, there are some occasions where you won't be able to factorize a quadratic equation, and that's where you want to use a formula to find the two results of the equation. That formula is; when we have a quadratic which is ax squared plus bx plus c equals zero, the results are equal to negative b plus or minus the square root of b squared minus four ac, all divided by two a. So we'll use this equation here again to show that that's true, but obviously if you've had this equation, you would normally factorize this equation and it would be quicker than using this formula. But if we do use this equation again, we can see that x is equal to negative b, well, in this case, b is negative six, so negative b is six - plus or minus the square root of b squared, minus six squared is thirty-six, minus four times a; well a is the number in front of the x squared, in this case that's essentially one, even though it's not written; and c, which is five. And we divide all of that by two a, so this is equal to six plus or minus the square root of thirty-six minus twenty, all over two, which is equal to six plus or minus the square root of sixteen, over two, which is three plus or minus four over two, which is two, and therefore x is equal to three plus two which is five, or three minus two which is one. So you can see that we have the same result that we had here. There is only one possibility when this equation won't work, so, if you can't factorize it, you can use this equation; if you can't use this equation, then there is reason you can't use it, and that will be because b squared is less than four ac, and when that's the case, it simply means that there are no real results to this equation, and you cannot find numbers like five or one, which would usually solve the equation. And that is how to solve quadratic equations. .