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# CA Algebra I: Quadratic roots

## Video transcript

or on problem 58 the graph of the equation y is equal to x squared minus 3x minus 4 shown below fair enough for what value or values of X is y equal to 0 so they're saying essentially saying is when does this here equal 0 right they want to know when does y equal 0 so what values of X does it happen that does that happen and we could factor this and solve for the roots but they drew us the graph so let's just expect it so when does y equal 0 so let me draw the line of y is equal to 0 so that's right here let me draw it as a line y equals 0 that's y equals 0 right there right so what values of X makes y equal 0 if I'm I can see this properly it's when X is equal to negative 1 and when X is equal to 4 so X is equal to negative 1 or 4 and if we substitute either of these values into this right here we should get Y is equal to 0 and let's see the choices they have negative 1 & 4 yep sure enough negative 1 & 4 right there next question 5959 let me let me copy and paste it okay I have copied it let me I'll paste it below I'll do it right on top of this there you go 59 let me erase this stuff right here unrelated to this problem okay so they are asking pin right which best represents the graph of y is equal to minus x squared plus 3 so here am I just to get an intuition of what parabolas look like because these are all parabolas or the graph of a quadratic equation so if I had the graph of y is equal to x squared what does that look like well if I just let me just draw a quick and dirty x and y axis and I think you're familiar with what that looks like so if I were to just draw Y is equal to x squared that looks something like this it looks something like this it looks something like that I think you're familiar with it as you go because you're taking x squared you always get positive values right even if you have an negative numbers squared that still becomes a positive and it's symmetric around around the line X is equal to zero all right that's what the graph of y equals x squared now let me ask you a question what is the graph of y is equal to minus x squared let me do that down Y is equal to minus x squared so it's essentially the same thing as this graph but it's going to be the negative of whatever you get right whatever x value you so here x squared is always going to be positive right you square any real number you're gonna get a positive number here you square any real number this part right here becomes positive but then you take the negative of it so this is always going to be a negative number so at X is equal zero is still going to be there but regardless whether you go into the positive x-direction or the negative x-direction this is going to be positive when you put the negative sign it's going to become negative so the graph is going to look like this the graph is going to look like that I didn't draw that well let me give that another shot the graph is going to look like that it's essentially like the mirror image of this one if you were to reflect it on the x axis right this is y equal minus x squared so now you're pointing it down the u goes opens up downward and hopefully that makes a little bit of sense and now what happens if you do plus or minus 3 so what is y is equal to x squared plus 3 y is equal to x squared plus 3 not minus x squared plus 3 but it's x squared plus 3 so if you start with x squared now every Y value for every given X is just going to be 3 higher so it's just going to shift the graph up by 3 it's going to look like that so if you go from the x squared 2x squared plus 3 you're just shifting up by 3 similarly if you go from minus x squared to minus x squared plus 3 which is what they gave us in the problem you're just going to shift the graph up by 3 so I'll do that in this brown color so just going to take this graph which is minus x squared and you're gonna shift it up by 3 so it's gonna look something like this it's gonna look something like that so let's see out of all the choices they gave us it should be opening downward and it should have its y-intercept at at Y is equal to 3 all right if you put X is equal to 0 Y is equal to 3 so let's see what if this opening downwards so these two are the only two that are opening downwards and the y-intercept should be a 3 because we shifted it up by 3 so this is a choice B problem 60 which quadratic function when graphed has x-intercepts of 4 and minus 3 so X intercepts of 4 and minus 3 means that when you substitute X of either of these values into the equation you get Y is equal to 0 right because when y is equal to 0 you're at the x-intercepts this is when y equal to 0 so that's what they mean by x-intercepts so how do we set up an equation where if I put in one of these numbers I'm going to get 0 well if I make it the product of the X minus the first root and X minus the second root so X minus minus 3 is X plus 3 so think about it if you put 4 here for X you get 4 minus 4 which is 0 times 4 plus 3 7 so 0 times 7 is 0 so that works and then 4 minus 3 minus 3 minus 4 is minus 7 but then minus 3 plus 3 is a 0 so either of these when you substitute it into this expression you get 0 let's see which choice is that X minus 4 X minus 4 times X plus 3 halves the x-intercepts X X intercepts of 4 and minus 3 right this should be right oh you see they're they're being tricky right here so X plus 3 is there I see that in a couple of them right but I don't see the X minus 4 anywhere but that's because we could multiply this by any constant right because you know 0 times some number some cons it's still going to be 0 so if you look at this one right here 2x minus 8 2x minus 8 we could factor out a 2 that's the same thing as 2 times X minus 4 right so choice B is X plus 3 times X minus 4 times some constant 2 so choice B is our answer right if this is equal to 0 when X is equal to 4 minus 3 this sum any constant X minus 4 times X plus 3 that's still going to be equal to 0 right because if when X is equal to 4 this is going to be equal to 0 so 0 times anything times anything else is going to be 0 same thing with X is equal to minus 3 so they just put a 2 here that's a good problem made you realize that you could put a constant in there and it's a little tricky next problem next problem okay so they want to know let me copy and paste it how many times that the graph of y equal to x squared minus 2x plus 3 intersect the x axis so the easiest thing to because you know maybe it doesn't intersect the x-axis at all maybe well if you use a quadratic equation there are no real solutions so let's just apply the quadratic so the the the roots or the the times that I guess the X values that solve this equation right 2x squared minus 2x plus 3 is equal to 0 and these are the X values where you intersect the x axis and why do I say that because the x axis is the line y is equal to 0 so I set y equal to 0 and I get this and we know from the quadratic equation the solution to this is negative B let me do this in another color it's a negative B so minus minus 2 is 2 plus or minus the square root of B squared minus 2 squared is 4 minus 4 times a which is 2 2 times C times 3 all of that over 2a 2 times 2 which is 4 now and they don't want us to figure out the roots training this one oh how many times does it intersect the axis so let's think about this what happens under this radical sign if 4 times 2 times 3 is 24 so this becomes 2 plus or minus 4 minus 24 over 4 this is minus 20 right - 20s you end up with minus 20 under the radical sign and we know if we're dealing with real numbers we want real solutions you can't take the square root of minus 20 so this actually has no solutions or another way to put it is there is no X values where Y is equal to zero or another way to put it is this never does intersect the x-axis so it's a none and the the the what gave that away was the fact that when you apply the quadratic equation you get a negative number under the radical sign so we're dealing with real numbers you there's no no answer there next question next question 6262 an object that is projected straight down or others good this projectile motion is predicted straight downward with the initial velocity V feet per second travel the distance of s V times T plus 16 T squared where T equals time in seconds if pheromone is standing on a balcony 84 feet above the ground and throws a penny straight down with an initial velocity of 10 feet per second and how many seconds will it reach the ground okay in hominy so he's 80 feet above 84 feet above the ground let's draw a diagram so that's he's 84 feet this is 84 feet above the ground and says how many seconds will it reach the ground so we essentially want to know how many seconds will it take it to travel s his distance right so s is equal to 84 feet it has to go down 84 feet so let's see if we can figure this out now this is this is something that might be a little bit so how long does it take it to go 84 feet I guess is the best way to think about it so we say 84 is equal to velocity times the your initial velocity times time and your initial velocity is 10 feet per second so it's 10 feet per second everything is in feet I think right everything is V feet per second right initial velocity of V feet per second so 10 feet per second times time I just substituted what they gave us 10 for V + 16 T squared and now I just solve this quadratic that is a T right there so let me put everything on the same let me subtract 84 from both sides and I'll rearrange a little bit so you get 16 T squared plus 10 t minus 85 or is equal to zero all i did is i well i swapped the sides I put this I put these on the Left well let me just show you what I did I swapped the side so I made this 16 T squared plus 10t equals 84 right I just swapped them and then I subtracted 84 from both sides to get this and now we just have to solve when T equals zero so that's the first thing we could do is we could simplify this a little bit everything here is divisible by 2 so this is 8 T squared plus I'm just dividing both sides of this equation by 2 plus 5t minus what minus 42 is equal to 0 and then we can let's see use a quadratic equation so what are the solutions T is equal to negative B so minus 5 plus or minus the square root of B squared so 25 minus 4 times a times 8 times C C is minus 42 so instead of x minus 42 let's put a plus here and du plus 42 just a negative times a negative is a positive all of that over 2 times a all right 2a is 16 so let's see where that gets me so I have T is equal to minus 5 plus or minus the square root what is this 25 plus C 4 times 8 times 42 that's 4 times a that's 32 times 42 32 times 42 see 2 times 32 is 64 put a 0 4 times 2 is 8 4 times 3 is 12 all right so you get you end up we get C 4 14 3 & 1 all right so this is 1344 we're gonna add this 25 here so let me see plus 13 44 all of that over 16 let's see 13 44 plus 25 so it's minus 5 plus or minus the square root of this 1313 69 over 16 and actually I don't know what the square root of 13 69 is let me get the calculator let me open it up give me one second accessories calculator all right so 13 69 69 37 look at that okay so it's minus 5 plus or minus 37 that's the square root of 13 69 so minus 5 plus or minus 37 over 16 is equal to the time now we don't have to worry about the minus coz that's gonna give us a negative number minus 5 minus 37 over 16 we don't want a negative time we want a positive time so let's just do the positive so minus 5 plus 37 let's say -5 plus 37 over 16 so that's 32 over 16 it equals to 2 seconds and that's choice a anyway see in the next video