hello this is John from TC math academy and what we're going to be doing in this video is practicing with the quadratic formula and of course the quadratic formula is a Formula that allows us to solve quadratic equations so we're going to solve this quadratic equation using a quadratic formula but let me ask you is there other techniques that we could use to solve this quadratic equation in other words you might be looking at this equation you're saying you know what I don't need to use the quadratic formula I could solve this in an easier method now if you think you could just solve this using a different technique go right ahead and attempt to do that either way go ahead and put your solutions to this equation if you know how to do or solve this particular quadratic equation put that into the comment section I'm going to show you the correct answer in just one second and then we're going to have a discussion on just in general quadratic equations and I'm going to solve this quadratic equation using the quadratic formula okay so this is going to be good practice for those of you out there that need to master the quadratic formula and if you're taking any sort of algebra this is an absolutely must know concept and skill also if you need math help with the course you're taking test prep or homeschooling make sure to check out my math help program at tcmathacademy.com you can find a link to that in the description below and if this video helps you out don't forget to like And subscribe as that definitely helps me out okay so let's go take a look at the answer here we have x squared minus 4X is equal to eight what is the solutions to this quadratic equation well here they are right here x is equal to 2 plus or minus 2 square root of three so this is what you should have and this is right here is actually two solutions so one is two plus two times the square root of three so this would be one solution and the other solution would be uh to minus 2 times the square root of 3 but kind of use this shorthand right here this plus or minus so we don't have to kind of you know be a little bit redundant here with uh writing all this out so this is the most uh correct answer now some of you may have gotten pretty close but you didn't maybe finish simplifying your results either way if you're going to do this particular equation in an algebra class or some sort of math class this would be considered the most correct answer okay so how'd you do well if you got this right that is very very good matter of fact I'm going to give you a nice little happy face and a plus plus a 110 percent and multiple Stars so you can celebrate your knowledge and skill with the quad quadratic formula and quadratic equations because this particular equation in fact you do need to use a quadratic formula to solve this particular equation now there is some other techniques you can use to solve this but let's just go ahead and start talking about this right now okay so here is the problem we have x squared minus 4X is equal to 8. so of course I have some things here that you want to be thinking about anytime you are faced with a quadratic equation so first of all what is a quadratic equation well it's what we call a second degree polynomial equation so uh here we have an x squared now some quadratic equations don't have an X term okay and of course if this variable is y uh you know all these variables are just you know be this would be y squared and this would be 4y so don't let the x squared you know kind of like confuse you it could be any variable but any variable squared we're talking about a polynomial squared where the highest power of in the equation is two okay so when you recognize you have this type of equation of course this is an equation the first thing you need to say is okay this is a quadratic equation so there will be two solutions okay so this little 2 up here indicates when when you have a polynomial that you're dealing with a second degree polynomial and there's something called the fundamental theorem of algebra basically says that listen whatever the highest power of your polynomial is that's how many solutions you're going to have now what type of solutions well here we just don't know that right now we're just kind of looking at the equation you could have two real number Solutions or you can have complex or imaginary number Solutions we don't know but we do know we're going to have two solutions okay so what else do we know about quadratic equations well one method that you need to be aware of is that you could possibly take the square root of both sides now that works out when you have a situation where you have a square on either side of the equation something like x squared is equal to 16. you see here this is a quadratic equation second degree of polynomial but we don't have an X term so in these type of situations we could just simply take the square root of both sides so this is like C super awesome so X is equal to positive negative 4 that would be the solution to that equation there all right now in this particular case this equation we're not it doesn't it has an X term so we're not going to be able to do this right now so we can't do this we know this has a two two solutions so what we want to uh kind of look at next is can we Factor this right in other words if I move this 8 over on this side I have a quadratic trinomial x squared minus 4X minus 8 is equal to zero could I factor this into two binomials okay now this is a method that you should attempt to do now you're not going to know whether you can Factor this unless you kind of move this 8 over and you attempt to factor this but in this case this trinomial Guild cannot Factor this is what we call a prime trinomial so we cannot Factor uh this trinomial now if you could that you know you kind of want to use factoring but if you cannot factor and you can't take the square to both sides well what are we left with well we have the quadrant radic formula to the rescue okay the quadratic formula will solve any quadratic equation so we anytime uh you know we can't use any of these other techniques we can always go to the quadratic formula and there is another technique called completing the square which is kind of like the long version of the quadratic formula you need to know this as well but from a practical standpoint really if you can't Factor if you can't take the square both sides you're just going to fall back on the quadratic formula okay so in the beginning of this video I kind of stated and said hey if you think you can solve this use another another technique go ahead and try okay probably must do probably attempted to factor this and hopefully you know you know how to factor like you know I can't Factor this so I do have to go to the quadratic formula okay so what is the quadratic formula well here it is right here so just in case you forgot what it looks like this is is basically it all right now let me explain this here real quick so here is a quadratic equation uh in what we call standard form all right so this is just a general quadratic equation a x squared plus BX plus C A is the coefficient here of the x squared term B is a coefficient I these are numbers right in front of these variables and then C is just a number all by itself so when we know the a b and c values we plug it into the actual quadratic formula which is the following right X is equal to minus B plus or minus the square root of B squared minus 4AC all over 2A this formula here I would commit to your long-term memory now there are a lot of formulas in algebra and Mathematics and that's why you take notes because you can't possibly memorize all formulas but this one here you uh I'm going to suggest that you do kind of memorize this in other words you know reference your notes just to make sure you have the correct formula but as you practice this try to you know actually remember this so you can just kind of draw upon it at any time okay so we're going to be using the quadratic formula to solve this quadratic equation and let's go down here and take a look at the actual problem so we have x squared minus 4X is equal to eight now here this the way the equation is written out it's not in standard form okay now standard form in other words we have to get an equation where everything is on one side of the equation and then we have it equal to zero okay now here we don't have everything on one side of the equation we have this 8 on this side so we have to move this 8 over here so we'll subtract a from both sides of the equation and now we have x squared minus 4X minus 8 and you're going from highest to lowest power right so you have x squared your X and then your number so this is our ax squared plus B X plus C format okay so in other words right here you can see it's in this form right here okay all right so at this point all we could do is go ahead and identify the coefficients which are the numbers in front of these variable terms so right here there is a one okay and one right there so our a is equal to one here there's a negative 4 right here that's the negative so our B is going to be equal to negative 4 and then our constant here is negative eight so that's going to be our C value so we have our a b and c values so I've kind of set all this up for you so if you're like okay I understand well that's great okay so if you didn't get this problem right go ahead and plug in these respective values for a b and c and two the quadratic formula and then simplify this thing and see if you can get the right answer which I kind of showed you in the beginning of this video because um this just kind of doing the math here gives a lot of students trouble okay so I want to give you an opportunity to practice this if you were kind of confused uh in or you know the beginning of this video or the problem you're quite sure what to do so see if you can do this part because if you can't do this part then you know obviously you're going to need to be able to practice using a formula which is this uh in the whole idea of this video all right so let's go ahead and do this right now okay so here we have our a b and c values so here's the quadratic formula so we're going to have to be super careful to replace each one of these variables with these values so here's B so we have minus B so I have to replace this B with a negative 4. I'm also going to have to replace this B with the negative 4 and then of course I have a I don't have to replace that with one and then C I'll have to replace that with negative eight and then here a is with one now when you're plugging your values in to the quadratic formula you want to use parentheses okay so in other words here I have minus B use parentheses to plug in your values so this B is negative 4 so this you're going to have a negative parenthesis negative four all right so let me show you all this work right here so this is minus B so it's negative of a negative 4 plus or minus square root parentheses again right negative 4 squared minus 4AC all over 2A so the first um kind of part of using a quadratic formula after you after you've kind of identified your correct a b and c values is to be super super careful plugging in those values into the quadratic formula now a lot of students make mistakes they'll plug in a number they'll have the wrong sides and again if you use parentheses that can really help uh kind of reduce errors so before you start doing the math here double and triple check you know go back and say okay did I plug everything correct do I got my correct a b and c values double check triple check and then if you're like satisfied that yes indeed you plugged everything in incorrectly then from this point forward we can kind of start simplifying all of this so let's go ahead and get into this right now okay so first things first we'll go up here we have minus a minus four okay with course opposite of a negative four that'll be a positive four plus or minus so here is negative four squared negative four times negative four is positive 16. now this part right here all underneath this square root is what we call the discriminant and this is a kind of common error where a lot of students make mistakes with a quadratic formula so this minus 4AC part this minus if you turn this into a plus negative that will help kind of identify what sign this number will be so we have a negative times a positive times a negative so this is all going to end up being a positive value right so we know that 4 times 1 times negative or times eight it's just it's going to be we kind of just forget the signs here because we know the final answer is going to be positive so we just go four times one times eight which of course is going to be 32 all over 2 times 1 which of course is 2. all right so you're just going to be working um uh you know all this math down step by step you don't want to take too many steps you know just in other words you don't want to go from all of this to everything all at once and then have one final thing you want to take only a one or two steps and continue to write so uh you and your teacher can keep track of what's going on all right so let's go ahead and continue uh the problem so we have four plus or minus right so now we're going to add these numbers underneath the square root 16 plus 32 is 48 so we have 4 plus minus square root of 48 all over 2. now if you had this as your answer I would give you a happy face okay yeah I might even give you an A minus maybe like a 93 so that's pretty good but here's the problem you have some work to do here this is not fully simplified but this is um you know an indication that you did plug in the correct values into the quadratic formula you know what you're doing but you're not done yet we actually got a decent amount of work to go so let's go ahead and continue to simplify uh this situation and namely I have this square root the square root of this 48 anytime you have the square root when you're dealing with the quadratic equation you need to see if you can simplify this radical and in this particular case we can all right so let's focus in on the square root of 48 so the square root of 48 is equal to the square root of 16 times 3 okay so you want to look for perfect square factors here now this is a whole another kind of conversation or skill set working with uh radicals and square roots and at this stage if you are completely kind of overwhelmed let me give you a couple suggestions one I'm going to direct you towards my algebra one course and my math health program also I have a ton of additional videos on my YouTube channel as well but make you know note of what you're not understanding okay because the same thing is going to come up over and over again when you're doing these problems all right so the square root of 48 we're looking for perfect square Factor so 48 is the same thing as 16 times 3 so the square root of 48 is the same thing as the square root of 16 times 3 and then I could break this big square root into two separate square roots so this would be the square root of 16 times the square root of 3 that is a property of radicals or property of square roots and uh now that we have a perfect square factor of 16 we can take the square root of 16 which of course is 4. so now we have 4 times the square root of 3 so the square root of 48 is equivalent to 4 times the square root of 3 and that is what we need to use to continue to simplify this problem okay so here is where we kind of left off we had 4 plus the square root of 48 all over 2. so we did all this work to simplify the square root of 48 into 4 square root of 3. so now we need to kind of um factor out some numbers here some greatest common factors and clean this up because you can see two we'll go into both these fours right here but just to make it super clear on how to kind of simplify this let's take a look at the numerator here so here I have a 4 and I have a 4 here and it's a this is plus or minus so I can factor out this greatest common factor four right so in other words if I take that 4 and multiply back in uh that would be 4 times 1 over 4 and then this 4 times that square root of 3 would be 4 square root of 3. so this is a factor so now I have 2 and down in my denominator so 4 of course is the same thing as two times two so I can cross cancel one two that'll leave me with a 2 right there okay up in the numerator so I'll take this 2 that'll go into that for twice so now I have 2 times 1 plus or minus the square root of 3 and then of course I can distribute back in take this 2 multiply back in two both of these terms right here and let's go ahead and wrap this up so this would be 2 times 1 which of course is 2 and 2 times the square root of 3. so this is the final answer okay so X is equal to 2 plus or minus 2 times square root of three and again as I kind of indicated in the beginning of this video all quadratic equations have two solutions so just to be super clear the two solutions this plus or minus means one of the solutions is 2 plus the square root of three the other is 2 minus the square root of three so these are the two unique Solutions but we like to kind of use this shorthand here this plus or minus just to make it easy to write all this out okay so how'd you do well if you again if you did this correctly that's very very good but I would say this problem is a pretty easy problem when it comes to the quadratic formula so you know uh the whole point of this video was one just to practice quadratic formula two to kind of you know think about the quadratic formula in the bigger picture of quadratic equations and again if there is a skill whether it be factoring or working with square roots that you're not you know comfortable with or you're you know just not confident about you need to address this because this is not going away okay so hopefully this video helps you out if that's the case don't forget to like And subscribe and with that being said I definitely wish you all the best in your mathematics Adventures thank you for your time and have a great day