 So if you want a quick answer as to what the point of algebra is, you might say that the point of algebra is to be able to solve equations. And an important type of equation that we want to solve is called a quadratic equation. It's an equation, it's a polynomial equation with a square as its highest power. So here's a 30 second overview of everything you need to do in algebra. All of algebra can be viewed as a way of reducing any equation to one of a couple of standard types, which we can then apply an immediate reduction. So let x and y be algebraic expressions and be some number. Then what I have are the following main types of equations. So I have product equal to zero. If I have a product to see the expression, so that's equal to zero, then I know one of them must be equal to zero. Likewise I have quotient equal to zero. If I have a quotient of two expressions that's equal to zero, then I know that the numerator expression has to be zero and importantly the denominator expression can never be equal to zero. I have square equal to number. If I have the square of an expression equal to some number, then the expression itself is going to be plus or minus the square root of that number. And what that means is that if I have a quadratic equation, a quadratic expression, a polynomial that has a degree of two, I can solve it by factoring. However, in most cases a quadratic expression can't be factored. So in general factoring is a waste of time. I might, if I'm really really lucky and really really really really really patient, I might be able to write my quadratic expression in the form product equal to zero, but most quadratic equations can't be factored. So in general, trying to do factoring is going to be a waste of time. The other possibility is I can complete the square. And what that means is I'm going to do something and I'm going to rewrite my equation in this form x squared equals something and then I can get x equal plus or minus the square root of n. And the downside to this is this always works so well actually there's no downside to it. It always works. Well let's take a look at that. So let's try and solve this by completing the square x squared plus 8x plus 10 equals zero. Now to make our work easier what we'll do is we'll isolate our x squared and our x terms. And the reason for that is that it's easier to complete the square if we don't have this extra constant hanging around. So I don't really want that plus 10 here and again quick analysis of the left hand side. This is a sum of things so I can get rid of things. I can change things by subtracting one of the add-ins and in this case I'll subtract 10. And over on the left hand side plus 10 minus 10 this goes away. Over the right hand side I have minus 10 and there's my simplified form. So now I can complete the square here. So how do I complete the square? Well here I'm going to take half the coefficient of x and square it so I'm going to add 16. Except if I want to maintain an equation I have to add 16 to both sides. So on the right hand side I have 6 and the left hand side I have the square of x plus 4. So I have x plus 4 squared equals over the right hand side 6. And now I have square equals number and that's one of my types of equations so that tells me that I can take the square root of both sides x plus 4 is importantly this is plus or minus the square root of 6. And the reason is that the square root symbol always refers to the positive number whose square is the radicand but I do have the possibility that I may have the negative solution as well. So I have x plus 4 equals two different things and again on the left hand side a little bit of analysis goes a long way. I have a sum so I can change a sum by subtracting the add end. And this is a really challenging task I'll subtract 4 from both sides and I have x equals plus or minus square root of 6 minus 4. Now we should always view this as two solutions depending on whether I use the positive square root of 6 or the negative square root of 6. So I should say the solutions are square root of 6 minus 4 that's my positive square root or if I use the negative square root that's negative root 6 minus 4 and there are my two solutions. What about if I have something like 4x squared minus 20x plus 1 equals zero? Well as before I do want to isolate the x squared and the x terms. So again over on the left hand side I have a sum I'm adding one and I can change a sum by subtracting I'll subtract one and that gets rid of the constant on the left and I have this and well I've only ever completed the square when I have a coefficient of one of the x squared term. It is possible to do so if you have a non non if you have another constant but it's a lot easier if the coefficient of x squared is one. So let's see I can divide everything by four and if I do that the coefficient of my x squared term is going to be one my coefficient of my x term 20 over four my constant one over five. So let's go ahead and do that I'll divide all of my terms by four and I'll do a little bit of simplification and now I want to complete the square over on the left hand side I have x squared minus 5x and I can complete the square by adding the square of half that coefficient so five halves squared that's 25 fourths I'll add 25 fourths to both sides and so now the left hand side is going to be the square of x minus five halves so over on the left I have x minus five halves quantity squared over on the right negative one quarter plus 25 fourths is going to be 24 fourths otherwise known as six and importantly now I have this equation in the form something squared equals number so I can take the square root of both sides again I do have to use that plus or minus and I have x minus five halves equals plus or minus square root of six and I want to solve this equation for x so now x is part of a subtraction I am subtracting five halves so I can undo that by adding five halves and my solutions are going to be positive square root of six plus my five halves or negative square root of six plus five halves and there are my two solutions