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# Introduction to the quadratic equation

## Video transcript

welcome to the presentation on using the quadratic equation so the quadratic equation it sounds like something very complicated and when you actually first see the quadratic equation you'll say well not only does it sound like something complicated but it is something complicated but hopefully you'll see over the course of this presentation that it's actually not hard to use and in a future presentation I'll actually show you how it was derived so in general you've already learned how to factor a second degree equation you've learned that if I had say x squared minus X mmm x squared minus X let's make that a minus ignoring that minus 6 equals 0 if I have this equation x squared minus X minus 6 equals 0 that you could factor that as X minus 3 and X plus 2 equals 0 which either means that X minus 3 equals 0 or X plus 2 equals 0 so X minus 3 equals 0 or X plus 2 equals 0 so x equals 3 or negative 2 and a graphical representation of this would be if I had this is my very badly drawn graph and if I had the function f of X is equal to x squared minus X minus 6 so this axis is the f of X axis you might be more familiar with the y axis and for these for the purpose of this type of problem it doesn't matter and this is the x axis and if I were to graph this equation x squared minus X minus 6 it'll look something like this I'll be it like this is f of X equals minus 6 and the graph will kind of do something like this I'll go up it'll keep going up in that direction and I know it goes through minus 6 because when x equals 0 f of X is equal to minus 6 so I know goes through this point and I know that when f of X is equal to 0 so f of X is equal to 0 along the X ax right right because this is 1 this is 0 this is negative 1 so this is where f of X is equal to 0 along this x-axis right and we know it equals 0 at the point X is equal to 3 and X is equal to minus 2 that's actually what we solved here maybe when we were doing the factoring problems you didn't realize kind of graphically what we were doing but if we said that f of X is equal to this function we're setting that equal to 0 so we're saying this function when does this function equal to 0 when is it equal to 0 well it's equal to 0 at these points right because look this is where f of X is equal to 0 and then what we were doing when we solve this by factoring is we figured out the x values that made f of X is equal to 0 which is these two points and just a little terminology these are also called the zeroes or the roots of f of X let's review that a little bit so if I had something like f of X is equal to x squared plus 4x plus 4 and I asked you where are the zeros or the roots of f of X that's the same thing as saying where does f of X intersect the x axis and it intersects the x axis when f of X is equal to 0 right if you think about that graph I had just drawn so let's say if f of X is equal to 0 then we could just say 0 is equal to x squared plus 4x plus 4 and we know we could just factor that that's X plus 2 times X plus 2 and we know that 0 it's equal to 0 at x equals -2 x equals minus 2 well that's that's a little x equals minus 2 so now we know how to find the zeros when the the actual equation is easy to factor but let's do a situation where the equation is actually not so easy to factor let's say we had f of X is equal to minus 10 x squared minus 9x plus 1 well when I look at this even if I were divided by 10 I would get some fractions here it's very hard to imagine factoring this quadratic and that's why it's actually called the quadratic equation or this second-degree polynomial but let's set it so we're trying to solve this because we want to find out where N equals 0 minus 10 x squared minus 9x plus 1 we want to find out what X values make this equation equal to 0 and here we can use a tool called a quadratic equation and now I'm going to give you one of the few things in math that's probably good idea to memorize the quadratic equation says that the roots of a quadratic are equal to and let's say that the quadratic equation is a x squared plus BX plus C equals 0 so in this example a is minus 10 B is minus 9 and C is 1 the formula is the roots x equals negative b plus or minus the square root of b squared minus 4 times a times C all of that over 2a I know that looks complicated but the more you use it you'll see it's actually not that bad and this is a good idea to memorize so let's apply the quadratic equation to this equation that we just wrote down so I just said and look a is just the coefficient on the X term right B is just the coefficient a is the coefficient of the x squared term B is a coefficient on the X term and C is the constant so let's apply it to this equation what's B well B is negative 9 right we can see here B is negative 9 a is negative 10 C is 1 right so if B is negative 9 so let's say that's negative negative 9 plus or minus the square root of negative 9 squared well that's 81 minus 4 times a a is minus 10 minus 10 times C which is 1 I know this is messy but I hopefully you're understanding it and all of that over 2 times a well a is minus 10 so 2 times a is minus 20 right so let's simplify that negative times negative 9 that's positive 9 plus or minus the square root of 81 we have a negative 4 times a negative 10 this is a minus 10 and it's very messy I really apologize for that times 1 so negative 4 times negative 10 is 40 positive 40 positive 40 and then we have all of that over negative 20 well 81 plus 40 is 121 so this is 9 plus or minus the square root of 121 over minus 20 square root of 121 is 11 so I'll go here hoping you don't lose track of what I'm doing this is 9 plus or minus 11 over minus 20 and so if we did so if we said 9 plus 11 over minus 20 that is 9 plus 11 is 20 so this 20 over minus 20 which equals negative 1 so that's one root that's 9 plus because this is plus or minus and then the other root would be 9 minus 11 over negative 20 which equals my minus 2 / - 20 which equals 1 over 10 so that's the other root so if we were to graph this equation we would see that it actually intersects the x-axis or f of X equals 0 at the point x equals negative 1 and x equals 1/10 I'm going to do a lot more examples in part two because I think if anything I might have just confused you with this one so I'll see you in the part two of using the quadratic equation