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Solving Quadratic Equations

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already got his welcome back to the hiset math lesson two talking about solving quadratic equations so we have three objectives today is going to be to understand how do I identify a quadratic equation next to me to solve quadratic equations by completing the square factoring or using the quadratic formula completing the square in a previous lesson so that'll be review Vasa Factor before with other types of them equation with the quadratic formula. And then finally will write one variable quadratic equations represent context all right let's go to that one of the variables is to the second power like this one right here x squared anytime we see that that variable raise the second power you know it's a quadratic equation whether or not we have a coefficient present or not it's a quadratic equation example this first example doesn't have a coefficient right-leaning there's no number in front of the variable but it's a quadratic equation because we have a variable raised to the second power the next example we do have a Coke machine error code fishing before but that doesn't change the fact that it's so we're looking for is the variable racist I can power here we have another exam what do we have to look out for we do have the variable rates as pyrite but we have a little problem here here we have two variables guess you. We have a wine mixed in their same thing here we have a y in this case that's a red flag for us I have two different kinds of variables then it is not a quadratic equation you can have two of the same variables for example but they're both the same variables are both the letter x at first glance this one looks fine but then we see here until we have the Y variable makes in there and it's not that always has to be it could be if it's all y-variable. Do you think that's okay use two different kinds of variables in the same equation right they can use any variable they want but they have to stick to that same variable in the examples that we're going to see you know the most common bird alright so let's take a look at what standard form for a quadratic equation looks like all right this is the standard form a x squared plus BX plus C equals 0 what do these mean the x is just the variable in the equation in the first term end of the second term with the reason they have in a year and a be here it's a show that there can be a coefficient write the A and the B stand for you know any number that can be a coefficient hear the sea sense for a constant right constant is just a stand-alone number so any of these letters that they that be in the sea can be replaced by numbers right although you may not necessarily Alvin and the Beast breakfast the sea inlets going to be a constant but you can have x squared + x + 7 right on that could be an example to take a look at the ones they give us right to the first example down here is 4x squared + 8x + 12 okay so you are the coefficients so the coefficients are four and eight I got to go fishing is a number that's in front of her now the constant is 1212 is a constant because it is not in front of a variable right it's just a stand-alone number that doesn't change so that's a constant here we have x squared - 10 + 24 is 10 though we don't see the number one in front of the x squared it's in five anytime a variable does not have a coefficient the implied coefficient is one the next coefficient is -10 in front of the X and then our constant is 24 or stand-alone number that doesn't change this is going to be the standard form in which you know the quadratic equations are going to so it says in the equation x squared minus 9 equals 0 coefficients are 10 and the constant is -9 in this case they don't even have a second term right they don't have a second you were here since there is no X in the equation I would have been our second trip so the coefficient of B is 0 if it's not if you don't have a variable at all if the axe is missing for the coefficient of 0 if the x is present with no number in the coefficient is 1 this is more of a rare case you know I rarely see stuff like this usually three equations all right click and drag the best answer to the designated place okay so - 12 what variables that represents the a the Bee or the sea in the quadratic equation what is a -12 what variable what letter what letter does the negative pole stand for remember back to the original standard form equation we saw there was a x squared plus BX what would the negative 12 represent the age of beer this year what do you guys think Google home something what are you guys doing what is the negative 12 represent a the beer the see nothing happens if you get it wrong so don't be I'm trying to share be yeah so this is -12 is going to be fancy all right now we're just down to a + b what is the a 1 14 right A B C we saw that on the previous slide where we had a x ^ 2 + BX + C remember that it'll be in alphabetical order Michael order when we look at the letters and it's in numerical order first term since for the first letter the second term to intuitively it set up so that the numbers of the letters match up there what is my a term what is my HR 2 where's my Beecher -10 94 we're just going in order think of a as 1 vs 2 and see us Alpha numerical order okay you were missing the song so why don't we start with the first what is the first turn what's the first from here hey is why he is one all right and I were missing be right we don't have the second term which would be would be represent and that's where we have the X variable right just ask to the first power so do you guys remember what we put in place of that if it's missing Euro Gyro hey Google right to remember if you just if you don't have a coefficient that's written for example this is have a coefficient that's written right there's no number written in front of the X so there's no efficient written then it's implied that it's one if there's no variable written meaning that we don't have an X here right there s trim is missing is no ex if the variable is missing in the coefficient of 0 alright I'll talk about rearranging the quadratic equation are equation in the standard format right we're going to need to rearrange it into the standard form so let's take a look from left to right top to bottom right it's on the top left of x squared equals 3x - 1 so I noticed that you know instead of moving the x squared over the right they moved the second and third term over to the left right and that's because we want to have all of our terms on the left hand side of the equation and a zero on the right hand side of the equation let's see how we have a zero at the end so that's why they moved to the left so we subtract 3x from both sides now it's on the left side once both sides now it's on the left figuring out what its operation is and then doing the opposite to both sides in order to switch it from one side to the other so now they have it in standard form x squared minus 3x plus 1 do they have the quotations 1 - 3 and 1 perfect you they need to distribute first so they need to distribute the two into the parentheses so then they multiply 2 * W squared + 2 * 2 w once they do that they have to subtract the five from both sides so that's going to give them to W squared - 4 W and then once you subtract that five from both sides then they're going to have it on the left this is there equation and then again the coefficients are to -4 and their constant is -5 all right here same situation as the one about right the last one but they need to do is going to distribute busy so they multiply z x e. D square z x one that's the so then they end up with Z squared minus and they subtract 3 from both sides in order to move it to the left and have a zero on the right here there are these squared - 3 equals 0 and then hear the code for today we're going to take care of that if it's on Center form will put it in send it for my this is all just rearranging the numbers right we're not changing the value we're just changing the form it's kind of like if you know let's say you own like a vending machine right and when you get the money it's going to be in all different kinds of denominations right assuming that the vending machine give change you know you're going to have singles may have five Sons funnies read me have different denominations of coins right assuming they machine steps all coins and then at the very least is you know all the money is going to be intermixed you at least want to separate each denomination into a pile and then stack it from smallest to highest right and you do the same with the coins in order for you to have it in an organized manner so that you can use your money you no more easily right you know if you have all your money mixed up you probably don't even have any idea how much you have total organized denominations together smallest to largest bill aren't you back then you can easily access your money know how much you have done what you have left all of that so that's what we're doing with these equations are here we are changing the form but not the value to make it more manageable for us easier to work with Okay move on to a different topic let's talk about factoring quadratic equations so we've factored quite a bit in the past right now we're just factoring something different so it's read a little paragraph it says the process for solving quadratic equations is similar to the process of solving linear equations first of all the trim is the left side of the equation so that it is in standard form hopefully they provide the equation in standard form if not no worries we can take care of that right what's a quadratic standard form solve for the unknown variable equations can be solved using a method called Factory you know about that message right this is going to come real and he's taking a picture of this right here the fact that they have the factorization already set up depending on how the equation looks is super helpful right normally you know a person wouldn't have this right they won't be able to look at this chart at the four different types of equations that are provided to us we will just have to look at the numbers and look at the signs that have to determine when we factor which number is going to take the negative sign with something positive sign but again having this chance to take a picture of it you can grab it from the video Once all right so take a look at the steps right so we want to determine the factors of the constant term okay so which one's the constant term let me ask you guys what variable is represent represent which one is it which variable represents the cost is it April constant be for constant or C4 yeah it is you know coincidental right it just so happens that the third term is the constant right and the third letter in the alphabet is also see so it works right as it's not because it's the word con but it seemed right because the first two are coefficients and that's represented by a and be right so it's just you know more of a mere coincidence but it's let us take a moment to look over data graph that we have and then we're going to review it alright so let's see of a constant term of the quadratic equation the car seat and then it says whose sum is the coefficient of the second Factor when we have the factors it has some more what's 5 to the third term and combined to the second term right that's what has to happen in order for us to be able to fax it in some of the more and some of the examples that we looked at in the next couple of life but keep that in mind and combined to the when I see combined you know I'm using that term on purpose instead of using ad because you know we may have a negative term right in our factors so we have to keep remind right we we could have a negative term all right let's go on to the next slide. so says the quadratic equation x squared + 8x + 15 is a type-a equation all right so you know there's two plus signs there for a while so here's what I mean by it the third term is 15 second term is 8 so we need two numbers in here that multiply to the and add up to the 2nd right or combines the 2nd so the third term is 15 so 5 + 35 * 3 + 15 5 + 3 is at 8 so it work here they give you the example right so the sea is the third term they have to multiply to 15 1/2 to add or combine to be which is the second rent all right now we're giving an example of a type equation that means we have a plus sign between the first and second term and a minus sign between the second and third so that means when I opened up my print the Seas I'm going to need a plus in the first pregnancy and a minus in the second trim so I need two numbers that multiply out to -14 all right we'll positive 7 + -2 can do that right awesome 7 - 2 million - 14 now I also need them to combined and give me five rent so I can take those same numbers that positive 7 and -2 combined them and I get a 5 so that works to write to hear my factor x + 7 x - 2 take a look at a type-c all right this is when we have to minus in our original quadratic equation we have a - between the first and second trim and a minus between the second and third term right I need two numbers that multiply to -12 and combined -4 to hear they are I have a -6 + a + a 2 - 6 * 2 - 12 so that works -6 + 2 is me - 4 so that works as well I'm filing a pig look at the forest type of equation right this is when we have a negative a subtraction and addition in that order so you know we have a - between the first and the second term and an addition between the second and third term so we're going to do in this case is we're going to put two parentheses both containing a minus sign right and the minus and the negative they're interchangeable in the way that they function right subtraction and negatives function the same way so if I have a -4 * + -2 that's going to give me a positive eighth right so that works you can see my third term -4 + -2 positive 8 now -4 + -2 give me a -6 so that works for my second trip so we're all good so again helpful. picture of this right yet another chance to take a picture just to know you know do I have two plus signs minus signs or one of each and if I have one of each and what order do I present all right okay so if it's not in standard form we're going to put it in standard form write that's going to be the first and foremost thing right so this equation is not in standard form write we're supposed to have a 0 on the right so my first order of business is putting it in standard form the way that I do that because I add 6 to both sides and even say why are we adding 6 to both sides the reason we're doing it is because we're trying to get the -6 the left side of the equation and the way that we do that is by doing the opposite operation of what is currently occurring so what's happening right now subtraction what is the opposite addition and where do I do it to both sides so that's what's why and where we do what we do so now it's in standard form write x squared + 7x + 6 this is a Type A quadratic equation right meaning it's the first exam prep I'm going to put two plus signs in here right and I need to pick two numbers that multiply to 6 and add to seven rate six in one can do that for me write 6 * 1 + 6 + 6 + 1 + 7 so that works out beautifully all right now that it's factored I can solve for x right that's something that we haven't been doing as of recent right most what time will we do is you know we simplify we were Deuce that's the same thing we put you know equations into its standard form you know we Factor equations you know and we really haven't been solving for x as much right we've been evaluating for Expressions right that's not the solving for x value in expression that's not the same thing right we've done that recently evaluate an expression that's when they tell us what x equals and then we solve for that right before exit cell where were Robert we're just plug it in and solving the problem oh here here is a situation where we actually find out what x equals an we do it by taking the two factors and send them equal 20 you can see how in this line both of them are equal to 0 right or what they do in the next step is a separate the two facts right and the two factors are each separately set so that they are equal to zero do they have x + 6 equal 0 and then on the other side to have x + 1 equals 0 all you got to do is offer X and this is a one-step algebraic equation all you do is you look at the left hand side of your equation you have an x + 6 what do you want you want the ex how are you going to do it you're going to isolated by doing the opposite operation to both sides of the equation so that means I look at what's going on so what is the opposite of addition subtraction and where do I do it both sides of the equation so now I have six now I can shift my focus to the other fancy right I have x + 1 what do I want I want X isolated how am I going to do it by doing the opposite directions to both sides of the equation I see that there's a plus one going on what is the opposite subtraction where do I do it both sides of the equation 3x equals -1 wait how can x equal to different things right Axis one variable how can one variable have two answers a question the reason we have two answers is because both of these numbers make the equation true meaning if I were to take -6 and plug it back into that original equation that we're looking at one slide ago I would get zero write this equation is equal to 0 so it would give me zero now if I were to take -1 and plug it back into that original equation that we're looking out on the previous page I would also get zero meaning that both of these numbers both - 6 + -1 solve the equation their solution is to solution bright you know I guess it's kind of like the same way I'm thinking of a scenario little restaurant right and and they bring you a dish right but they did something wrong with the dish either let's see the wrong order you ordered a pizza they brought you a hamburger right there's two solutions they can either give you your money back or they can bring you the correct order right shoe solutions for the same situation kind of like what we have going on here to solutions for the same equation okay let's see we have here so it says when solving for x in quadratic equations the X values are called solutions to the equation we already know that in the last example x equals if 6 + x equals negative 1 are solutions to the quadratic equation rate we also know that solution values are also called zeros because they are also the x-intercept of the quadratic function y equals 0 okay look at this there for - 6, 0 + - 10 are the zeros of a quadratic function okay so what does that mean they aren't the zeros of a quadratic function do you know anything by that statement will not really right because we haven't discussed it yet but what they mean by this is if we were to graft this for y equals x squared + 7x + 6 + is Zero's as a column is going to be where this function crosses the x-axis it's going to CrossFit at the -6 and it's going to cross it at the negative one sounds familiar right we had something similar a blessings ago we're talkin about linear equations right and we also talked about systems of linear equations that's when we solve for 2 linear equations and what are the ways that we did that was if we had a grass we look at where they were intersecting where they intersected that was a solution right that's similar to this right now we're also dealing with a different type of equation write a quadratic equation not a linear equation but they're both equations and another thing they have in common is that their Solutions as are called can be found on a right on the linear equation the solution to a system of linear equations is where the lines cross here it's where it crosses the x-axis right and this problem right here and it's the zeros right so some similarities different concepts but some overlap right between the two and here we go as promised right I told you guys it was going to look like here it is this is the you I was talking about you were not true in the whole quadratic outside coordinate plane right we're looking at the left side of the Court right the right side is missing here but again the reason you're not showing it is because all of the action is happening on the left so that's why they don't include the right hand side just so you guys have an idea of what we're looking at here this is what I meant by the zeroes everyone can see those dark black horizontal line with the X here right that's the x axis this red function that we're looking at that's our equation when it's bro and those zeros were talking about are right here at the -6 and the -1 point right that is where the red or the red function is crossing the x-axis those are the zeros and again those zeros are the solution that equation meaning if I plug in -6 into this equation I'm going to get zero if I plug in negative 12 this equation I'm also going to get hence the name zeros all right looping back to the beginning right just a different example that right but we're doing this for repetition and being familiar with it my problem here is this equation in it isn't in standard form so I need to take care of that right I need to subtract 10 from both sides and the question is why are you subtracting 10 from both sides why because I want to get it into standard form okay and why do you subtract 10 to get into Center for okay question this 10 is a positive 10 right and again A+ functions the same way that addition to the opposite of that would be 10 or a -10 that's why I say I'm subtracting both sides in order to maintain balance that's why now I have my quadratic equation in standard form beautiful I love that here we have the telling us there's a type-c quadratic equation right you looked at the four different types plus plus minus minus plus minus minus plus right those are the four scenarios you're have a minus minus scenario that means when I factory can I go to open up my friend that sees I'm going to need one princi with a minus another Fancy with a plus let's see what happens during the next week so take a look at how this all cracks okay you see how they said both of these equal 20 are both set equal to zero solve for x so again by doing the opposite operation so what they do the opposite add 5 to both sides so now we have our solution we have 5 + -2 what is another word for these Solutions are going to refer to them Russell called what what's the other word that was used decide Solutions name of a number zero product property zeros zeros okay so again if we have another quadratic equation right notice of shape right you shaped and again these all look the same because these are both different looking mostly on the right hand side of the y axis but anyways I guess you're are zeros the -2 and the 5 the Y intercept is 0 and you can go to plug in these numbers -2 or positive I've been to the original page can it says when solving for x in quadratic equations that are called solution we already knew that so here we have third exam luckily this one is already in standard form 3 all we need to do is Factor open up two parentheses two numbers that multiply to 4 and add 25 those numbers 4 and 1/4 * 144 + 1 is 5 separate equal 20 a song subtract 4 from both sides look at the one on the right I have an Xbox one I'm going to subtract 1 from both sides so 84 - 1 are both solutions to the quadratic equation or zeros like these are the zeros all right what's a German text you I need to do is open up and I'm going to see my ex so what two numbers multiply to 9 and combine at 8 negative 9 in Watts going to be the other way around it's going to be positive time is because is 9 - 1 I need a positive B and a 9 x -1 gives me the negative now what each of those people all right now I can solve for x right so your factors are different then your your ex. You noticed what happened inverted okay are the steps same steps that we did right for your ex's first then take a moment to think and figure out what you're getting here if you need figure out which one is positive which one's negative this case has positive one is negative at the answers my factors third step is solving for x okay thank you again so I'm going to try it out on your own and then in the chat send me a letter by the fee once I have a few responses and if you don't know send me a response Dylan to say IDK or I don't know just so that I know that you don't know I don't know IDK not sure any response just so I know that you guys are engaged and if you're still working on it and just hold off and send me a letter responsive time to work on it and then we'll get together just make sure when you're sending your answers your sending it to me everyone probably doesn't answers and if you're not sure I can just not sure any response that I know that play some beach all right so I got responses but I'm going to check on the rest of you and you guys can just tell me if you're working or I'm sure Adam are you still working on it or are you unsure about the Daniel Barrett Barry can hear you okay or are you not sure Daniel Daniel key are you still working on it or not true not working how about a cheater they're working on it or not Ormond Beach we're trying to Shana to work tonight or not sure okay thank you those for you could did respond well we audio is x squared - 7x excellent I think it's two numbers that multiply to 12 and out of 10 - 7 so I could try What cities were because if I multiply -4 * -3 I get a negative set both of these I'm going to use the same line in Zero's my answers okay cuz if little term of a quadratic equation has a coefficient of 0 different technique is used for scenarios we only have two charms the first and the third right results of the value for axial beginning as well as positive a negative value okay and the square root of a number you know there's two square roots take care of the equation that provided is x squared minus 49 equals 0 what they do here is the isolate the variable is latex the same way we always so the at 49 to go outside when you have squared equals 49 squared + 49x equals + -7 you may say why do we have two answers or why is positive why not well 7 * 7 equals 49 right that's the positive and negative 7th times -7 plus what equals 49 both positive 7 + negative 7 m squared give you 49 we have both of those as our answers and again we only see each other indicate positive negative 7 or answers all right so it's a sobbing equations using difference of squares all right so we've actually seen this in this before the difference of two squares is a method of fracturing that requires the presence of perfect squares all right so here are examples of perfect squares on the left hand side right and then these are the factors and this is how the expression Michelle alright then says determine the solution in a quadratic equation by using the difference okay so you're they have an exam right 9 x squared - 16 what they do is both arms in the expression are perfect squares the first term so what they do is they take the roots of each one so that's 3 x 4 the nine expert and for the 16 and they open up to parentheses right and if they have the same truck reacts and 4 + 3 x 4 does noticed that both sets of the front seat is the first charm in the original problem order Mac 3xs first because when we multiply it a 9 x shows up first so that's why I have a -16 right I have to be different here we go and then finally they sent both of them is equal to 0 and they saw this is a two-step problem I have to take care of the constant before subtraction I'm looking at the one the left recent traction and then the 3x by 3 in order to isolate the X on the right hand side actually PX or they have their solution bright - 4 / 3 + 4 / 30 all right so he has some quadratic equations can be factored in what is known as a perfect square of a single I know me so that means they're going to have the same the same terms same sign in both of the print C-SPAN perfect square of a binomial this is a single by Noma right one by Norview I can be able to squirt like this because they exact same thing what time does Factor the given quadratic equation using the perfect square method of a single binomial message interesting Lee enough you know there's many ways to factor is the GCF like greatest common factor of all three okay and I may find the solution Again by setting setting is this the same thing in both offenses -1 all right solving for x in quadratic equations that are called Situation all right okay I hear they're just showing function lyrics in green - 10 x axis at the -1.5 all right this one we saw before I we just saw this is going to be solved using square roots so we're going to send the 25 would you like to do with this let me get this right now I can do the square root I don't have a symbol of my keyboard and the √ 25 + 5 * 525 + -5 * 9 that's if you my answer okay I'll take a look at this one right here fantasies X I'm going to need two numbers that multiply 9 but add a + -6 these numbers in Hindi and this works because I - 3 * a negative 3 gives me a positive not negative 3 work perfectly equal to 0 my answer would be 3 right both of these are the same equal 20 I guess so okay talk about a different message so this one's called completing the square we do is we take the middle term / 2 in a square 7 are we going to do is going to send third term over to the right hand side of the equation so they do not buy subtract subtract 7 from both sides and then they just have the first and second term on the left hand side is what I was talking about you for sparing it 16 they're going to do is that chick that number they're going to add it back into the original equation take a 16 day out of two sides and then they can Factor this new trinomial right and you have no plans these are all three axles for rent and then you get an EIN it is a square root 4 - 3 and then finally are they are they sent these these equations equal to 3 the negative 3x the quadratic equation zeros open the first thing they're going to do the right hand side of the subtraction so they eat from both sides and now they have replacement set up like this is x squared 6 equals -8 both sides of the equation then we can go ahead and down Factory the trinomial on the left is 3 square root break it down into three call cracks in again you know this is just another way of doing right I would only resort to this method if you're not able to factor in the traditional method is the First Methodist numbers that multiply to 8 and -6 right this case with this problem that's pretty simple 1 + 4 + 2 I want to pick 4 + 2 + 4 + 53 + 6 you have to be a -4 + a negative to give me a positive b - 4 - 2 even when I add them give me a -6 opposite set of -4 - 2 I did the opposite to South Rex positive or positive once you once you become familiar with this even be able to just do it mentally because all you have to do is answer one or two questions what two numbers multiply to 8 that's a fresh but the answer is yes then you just do the opposite sign that's the answer and that's how you were doing okay all right what's this morning what are the only two numbers that multiply to match in this case to me weather what's the numbers Logan's 3 how do you get 9 think about the times table the whole numbers right of course we can do 4.5 + 2 now we can take a look at the second trim it's a - 10 okay so we're not going to change our numbers it's still 9 + 10 - 10 we're just going to change the sign - 1 + -9 give me - 10 Sonic doctors -9 - 1 if I do the opposite opposite of both negative now I'm going to go and speak to square feet yeah then I would have to Pakistan what is 25 x - 5 squared 18 square root 5 equals 4 x - 5 -4 the one on the left by add 5 to go sides I get back before tonight the one on the right if I have bicycle size ticket so I mean just take a look at that that's completing the square only necessary if you're not able to do it the traditional we're able to do it mentally with some of these right either really Elementary number of the 12 x 12 that. I'll be considered an elementary you know dealing with and addition and some of these numbers see how their relatively small but they're not double digit numbers at least the ones we have to move to fight you right ones we've seen so far they've all been 9 or smaller so we just asked ourselves what two numbers once we have that these two numbers combined as one once we determine the sign you do the opposite that's a solution are you going to that Papa okay this is unfortunately not all quadratic equations by factoring it's impossible to factor the quadratic equation anywhere you see a bee or see that's going to be replaced with the first second or third check via second trim his first term to use their turn to your place these letters with the first second interview song I only do this if if you can't fax it when we talk about the different ways of that if none of those work also take a look at it. what they do here an example of one that can't be identify the first second and third once they have it plugged in then Adam Saleh prank 49 - 2 * 2 - 7 + 2 root of 33/4 n - 7 - √ 33 / 7 subtract 8 from -7 you could also have a negative position might have a negative download Sprint what is the great thing about this is that can be used to solve equation no matter that are in the ocean and this isn't just any negative signs not to forget them that's the first thing don't forget you you know don't forget to take care of double negatives right that's another thing you know I think all those two stops the negative sign and a decimal sign super important I know how important the negative sign or decimal points anyone who's ever had an error in their bank account when I think about how much money you have if you're in bed if that negative sign wasn't there then you would have that money from the decimal point at some point is the difference between the dollar and dollars dollars thousand thousand hundred thousand important okay from the solution of the following quadratic functions using the quadratic formula so that we can see all the steps to begin with you want everyone to be able to see Pittsburgh us so here is the original equation are we happy you have 2x squared + 11x + 5 we need to do is we identify our A1c chart I'm going to read that over here what is my age what is my age Isaiah 11 he is mine think of it intuitively if you knew absolutely nothing what would your inseam for your back for you because it's set up in Sweden this is my a charm if you show me a jumpsuit what I'm going to do is I'm just going to plug these numbers again so here we go should be squared all right we have this is our original equation these are my different terms my abnc charm this is our original formula and this is my formula with my trims plugged in so now I can go in so let's start with solving under the square root sign so I'm going to take care of is going to be 11 Square so that means 11 * 11 cuz I can is it going to be 121 multiplication rent so these are all that what's that what are you going to check what is 4 * 2 * 12 40 take care of what are we going to get inside we're going to get what is it 121 81 so now we have 81 and numerator what are we going to get when we divide up at 4 where we can become 98 perfect we're going to have nine and again we got that because the √ 81 is not so what's the difference - 11 + - 9 what's -11 9/4 x 15 Stephanie is going to be this one so let's take care of the one on the left is what is -11 + 9 -3 / 4 now I got - 2 what simplified version one or two I don't supposed to be number one on the right - 11 - 9/4 what is it is 11 - 9 think about a bank account or what and what is that -5 these are my two answers - 5 and I'm going to stay on the iPad because we have one more to you with a quadratic is this woman with Baseball Song it has many components you know what I will say is that he is very quickly manageable you know within a couple of steps that the formulas given to us or we have to remember but I'm not sure I know that for example I know that the wonderful question I'm not sure just to be safe I would Memphis memory I think you know the song Robesonia this is a new problem x squared + 7x + 4 is my a is my a -7 I have my formula in front of me I have a negative be overlooked these already negative so what happens when I positive next East where Justin Bieber look up under the square root sign for the what is -7 49 and what is Wharton Street -24 okay so tell me what you want under the square root of 25 what is the square root of 25 well let me know when you have seven what is 7 + 5 all right to that's going to be d.c. you got here you can switch back life scenarios and word problems as the square of a number added to three times the same number is equal to what is so going to be translating the word format numbers for me the square of a number we don't know what that number is so use a variable most common variables X so x squared added that means Plus three times the same number three acts like as it's the same number of losses and variable is equal to equal sign 28 and again before I was asleep before you even go to YouTube for me that as a last resort I really want to have you guys lost his work is because one it's a lengthy formulas memorized you have to input a Trek into the formula and have a fun if you miss one that's a last resort solution let's take a look at the problem and see if maybe this is what we can do take a look at it first is what two numbers multiplied equal 28 other than 1 and 28 what two numbers multiplied what time is it Warren Sapp and was your question I was going to ask you what was your question that you were asking that what's the numbers give 28 find them you know I would have to have one negative but you can also give you three so what if I had a positive 7 and a negative or positive 7 + -4 give me - 28 when I'm in August 7th give me a positive numbers negative or positive -7 Palms natsumi positive for that the portals and only be a positive answer here it is second question is what is the opposite of and again you know I'm saying we're looking at this problem we saw that it's manageable are the recommendations to know the times tables let's take a look at how much longer is is because all right 21 inches squared we don't know what the width is right so use W4 website we don't know what it is so we just use a variable in this case w we don't know what the length is 1 inch longer than twice the width so now but I have now I can subtract 21 from both sides to get it into standard form this one makes sense to use the quadratic formula I can't look at this one context matters of the measurements the length of a rectangle we're going to set up do you know how much the wind is yes or no no use w number to message 10 is it Illinois you know how much the rent is all right you don't know how much you like to use w but I do know something about the I know something else I know that area uw-w 2 w x w w r next I can subtract 672 from both sides 950 w-where w - 72 now we can go over well I passed the sound the same Andrew Wommack but I can work it out Friesian and I'm just going to keep the variable X represent read messages negative be my B it was my Leesburg was -672 what's the name of this song La Musica now multiply 4 * 2/4 easy part right and then 8 * 672 what's a negative number 8 * -6 5336 all right and then we can go ahead and ask seems like a bit it becomes what we call a double negative to add it we're going to add 100 476 number what's the square root of 74 so I'm going to go ahead Reservoir the song White Horse mom negative 74 on the Sims 4 what is -10 + 74 16 like it over here -1 - 74 -84 -21 is it so now since we can't have a measurement here my wisc should be 16 right was more than twice that so so you guys can see all of the work I asked you know what the answer is no variable answers now we do know something they replace all of the where can the eight what did people think Mitsubishi W W2 form. is this the end of the year which is our standard form for a quadratic equation second son is song Beautiful now the final step you know what number one get into quadratic equation 73 solve the length of bugging in this is more testing related every question is worth the same answering 1 + 1 is the same as make sense to follow easy questions first and save the last you only need 40% it was I was thinking strategy was leave your time vinyl seats no problem here we have a positive positive have a minimum graph call the minimum don't have a lowest point but it won't have the highest point why won't it because it continues upwards of the positive Infinity there's no maximum on a positive right now the opposite of negative negative negative all right so another another aspect of the problem is called the axis of symmetry L'Occitane is is the point where you can win this case axis of symmetry -2 it is the x value of the maximum in this case so if we have intersex minimum or maximum line of symmetry oh really you know if if we have a blank sheet right like a British and we're going to put two points 1.000 - 40 - connect all three points using a lie let me know this is going to go like this as 2nd to 3rd and then this is going to continue. Era and then I know this has a maximum I know it's negative mirror image of left and right so this one has a minimum and then they also tell us where the axis of symmetry is didn't tell us about the second Point by the only we knew that we need at this point so we did a line up here and I think it was an earthquake any other point I just replicate the same line on the opposite side all right or if it was just a little review rabble is basically a U-shaped another part of the axis of symmetry zeros are also the solution where is the line of symmetry people look at what is it I'll even thought -7 + there goes the line all the way down it's at -7 basically the point is Ashlee Simpson all right this is the longest lesson lesson this is a standard form a x squared Andy quotations are numbers that show up to the left of letters or variables if there is no confirmation written you can assume that the purple sheep is one if the variable is missing Alright lyrics negative negative positive on the right hand side okay one of the other ways to practice of sharing the difference of square the other way is completing the square or maybe out right depending on the sign of the surgeon better now final and foolproof method is using the quadratic formula every single type of this is where we used nothing else work again again abmc and then finally all that was dealing with equations the type of graph of each piece is called its negative positive travels have a minimum yes I'm officially survived enough

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