 Let's do another question where it's a lot you know a fair bit more complicated or a little bit more complicated We're in general questions like this You know throw a lot of people off because they get really confused For some reason, you know, it does look it does look like quadratic function, but embedded inside it It's got something it's got another function, which is may or may not be a quadratic function Okay, you know you could use the quadratic formula to solve this by saying whatever the box is is equal to your quadratic formula We're going to simplify it down a little bit because I don't want to constantly write down x minus 2 over x, right? What I'm going to do is and again, that's one of the reasons you use the let statement To you know make life easier for yourself, right? You don't want to constantly write down, you know a bigger expression Then then you have to because every time you copy something down And another point every time you copy something down or look away from your piece of paper and you come back to it There's more probability of making errors, right? So what you want to do is simplify your expressions that you're working with as much as possible During during the work during the process that way you eliminate simple human error involvement Okay, so what we're going to do is you know let a different number different letter equal this right? Let's use that q for example q is one of the ones that you use in general or you or z or t whatever you want, right? Anything other than a b or c because we already have a b and c and anything other than an x because we already have an x and y in general is off limits because y is In mathematics, it's it's it has the it has the assumption that it's your f of x. Okay, so we're going to use q So we're going to say let So in our original expression wherever we see x minus 2 over x We're going to substitute it with q simplifies our expression right simplifies our equation So now we got negative 3 q squared plus 5 q plus 2 right and This guy we can factor again. This is something that you know You could factor using the four-step method you could grab the negative 3 bring it over because negative 6 two numbers that multiply to give you negative 6 and that to give you 5 is going to be Positive 6 and negative 1 you can use the four-step method and solve for this, right? We're going to use the quadratic formula and right now. I'm sticking with You know numbers that factor fairly easily because I don't feel like bringing up the calculator and punching in the numbers, right? I don't feel like carrying over the root symbols, you know Working with the with the radicals left and right right because whenever you get something like that if you're solving for it They're either going to say, you know solve to certain decimal places or Give exact values for if they say exact values What you have to do is keep everything in their root form in their irrational form But no decimals right if they ask you to round something up to a certain number of decimals So you're gonna have to round things up to a certain number of decimals right now We're gonna stay away from that because we're just learning the processes later on when we start, you know I'm gonna go through a series or When we finish all the different factoring techniques, we're gonna go into you know we're gonna do a whole bunch of complicated problems and During that process. We're gonna get into stuff where you know, you can't factor out simply right now this one We can't right, but we're still gonna use the quadratic formula for this You could use the four-step method and still use your substitution your let statement, right? But since we're on the quadratic formula, let's use the quadratic formula So what we're gonna have is q is equal to our quadratic formula So negative b 5 or b is 5 right if you want you could write out your a b and c right your a is negative 3 your b is 5 and your c is 2 And it's a good idea to do this just so you eliminate any you know human error or your mind playing tricks on So it's going to be negative b which is going to be negative 5 Plus or minus the square root of b squared, which is going to be 5 squared minus 4 Times negative 3 times 2 all of it divided by 2a which is 2 times negative 3 so right now what we're gonna have is negative 5 so Q is going to be equal to let's just solve this mentally right we've got negative 5 Plus 7 is going to be 2 to the oops I forgot the negative here. Don't forget this negative 6 coming down, right? These types of mistakes cost a lot of marks during exams, okay? So you should always be you know rechecking scanning over your work trying to eliminate these types of simple errors, right? So you got negative 5 plus 7 which is 2 2 divided by negative 6 is going to be negative 1 over 3 Right 2 over 6 is a third and it's negative right so it's going to be negative 1 over 3 That's our first answer for q the second answer is going to be negative 5 minus 7 is negative 12 negative 12 by divided by negative 6 is 2 So our two answers we got for solving this quadratic equation is negative a third and 2 Now we haven't finished this question yet because we weren't looking to solve for q q is something that we introduced in You know in the algorithm in the process of solving this equation, right to simplify our expression, right? We're trying to solve for x right because this would say solve for x so what we're going to do is resubstitute x minus 2 over x for q and solve for this and These guys we've talked about before when we're solving equations for a series 3. Hey, this is just cross multiplication Right, but one fraction equals to another fraction. So we're going to grab the x multiply it up here grab the 3 multiply it up here So 3 when it comes up here multiplies both of them. So it becomes 3 x minus 6 and This guy x comes up multiplies negative 1. It's just negative x now if you're solving this So x is going to be 6 divided by 4 which is 2 over 3 I'm sorry, which is 3 over 2, right? And that's one of our x values, right? What we're going to do is solve this one Cross multiply 2 is just 2 over 1. So you're just going to cross multiply this up cross multiply to 1 up, right? Fraction equals another fraction. So it's just going to be x minus 2 is equal to 2x So your answer here is going to be x is equal to negative 2 and those are our answers For our original problem for our original question, which was you know the expression that we had, you know What was it? Negative 3 bracket x minus 2 all over you know over x all square blah blah blah blah, right? Our original question was asking us to solve for the x value, right? Not to solve for qq was again something that we introduced into the process to make the expression simpler for us to solve So what we did was use the let statement to simplify things for us that way We could work with it crunch crunch crunch get down to this level and then Resubstitute whatever our let statement was for the x's and solve for the x's and get our answers to original question, okay? Now when you get the answers here You're not really done in generally you could be done depending on the question that you're being asked in you know in the exam in in the homework assignment or You know if you're working somewhere if you're you know if your work requires you to solve for quadratic equations And there are you know jobs in real life that you get where you have to do this type of work, right? If they in general they're more complicated, you know You run them through a computer and the computer spits out of your answer, right? Or the process spits out your answer your algorithm spits out your answer Sometimes you have to do things manually, right? It's a pretty good idea to go through the process and solve for them to get to the to get to the answers Make sure you know to do a double check because if you're doing anything in real life If there's money on the line or if there's you know Something else on the line that you you know you have you know your job depends on it or people's lives depends on it Or whatever it is if you're building something you better be checking your solutions, right? Checking your answers and that's what we're gonna do right now. We're gonna check both these answers So what we got is that's our original expression, right? And these are our answers we got for the x so what we're gonna do is check our answers, okay? And we've done some of the stuff in series 3a checking solutions, right? So what we're gonna do is just plug in x is equal to negative 2 We're gonna do this one x is equal to negative 2 on the left-hand side of the equation to see if it equals the right-hand side Equation now for us right now the question is easy the right-handed equation is just zero So right-handed equation is just zero Our left-hand equation is just going to be this guy with the x as equal to negative 2 sub d So what we're gonna have is So our left-hand side equation is negative So what we're gonna do is just simplify this cruncher to see if this side is equal to zero if it is then we know That answer is correct. So negative 2 minus plus 2 so negative 12 plus 10 plus 2. Well, that's just zero So right away we got Left-hand side of this equation when you plug in x is equal to negative 2 is equal to zero Right-hand side of the equation is equal to zero. So this works out. So we know this answer is correct. Okay? What we're gonna do is do it for this one as well So I'm not gonna bother writing down The original expression first because this one's gonna take us a little bit longer because there's fractions already And there's a nice little storm coming in so it's gonna start raining soon So I got a zoom out of here so the cowards and get wet, okay? So we're checking to see if x is equal to 3 over 2 which is 1.5 works in our original expression, right? Should we stay with fractions? Let's use the decimals. So 3 over 2 is 1.5, right? So what we got right now is going to be negative So this is going to be negative 3 1.5 minus 2 is going to be 0.5 0.5 divided by 1.5. I should have stuck with fractions. Let's stick with fractions with this, okay? So I made a mistake be careful with your mistakes I know this isn't gonna work out, but I made a mistake here, right? Because this isn't a half That's 1.5 minus 2 is negative a half So this guy should be negative that guy should be negative that guy's negative that guy's negative and that guy's negative So all of these become negative Right, so what we have here is this is negative 5 I personally make a lot of mistakes when I'm doing these types of problems. So be careful with this, okay? So all of those should be negative so What ends up being in the middle here is negative a third, right? So negative this guy's positive Negative 1 over 3 squared is 1 over 9 and you simplify this it becomes negative 3 here So it's a negative a third plus negative 5 over 3 plus 2 now these guys have the same common denominator and you can add these So negative a third plus negative 5 negative 1 plus negative 5 is negative 6. So this becomes negative 6 Over 3 plus 2 negative 6 divided by 3 is negative 2 Plus 2 so this side is equal to 0, which is what we wanted We wanted the right side to be equal to 0 2 so our check works out. So our answer X is equal to 3 over 2 is correct. So both our answers work out through the check and You know, this is sort of a fairly complicated not a fairly complicated But it's medium level question that you you would end up getting When it comes to solving something the quadratic formula and when it comes to checking it because the checking requires a fair bit of number crunching as well Okay, so, you know, this is using the let statement to simplify questions for you to simplify expressions for you to be able to crunch numbers and Solve for your quadratic equations or solve for any type of equations, right? Because our original question was not a quadratic equation We converted it into quadratic. It wasn't a quadratic polynomial We converted into quadratic polynomial We solved it as a quadratic and then we substituted back in what q was for the x's and then solve for the x values there, right? Okay, hopefully this all made sense. There's a fair bit of stuff here and we will definitely use Substitution and the let statement a lot further when we get into, you know, factoring and graphing polynomials and solving for equations