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CCSS.Math: , , , , , ,

I have an equation right here it's a second degree equation it's a quadratic and I know it's graph is going to be a parabola this was a review that means it looks something like this or it looks something like that because the coefficient on the x squared term here is positive and it's going to be an upward-opening parabola and I am curious about the vertex of this parabola and if I have an upward-opening parabola the vertex is going to be the minimum point if I had two downward-opening parabola then the vertex would be the maximum point so I'm really trying to find I'm really trying to find the x value if you could imagine the X I don't know where actually where this does intersect the x axis or if it does at all but I want to find the x value where this function takes on a minimum value now there's many ways to find a vertex probably the easiest there's a formula for it and we talk about where that comes from in multiple videos where the vertex of a parabola or the x coordinate of the vertex of the parabola so the x-coordinate of the vertex is just equal to negative B over 2a and that just the negative B you're just talking about the coefficient or B is the coefficient on the first degree term is on the coefficient on the X term and a is the coefficient on the x squared term so this is going to be equal to B is negative 20 so it's negative negative 20 over 2 times a over 2 times 5 over 2 times 5 well this is going to be equal to positive 20 over 10 which is equal to 2 and so to find the y-value of the vertex we just substitute back into the equation the Y value is going to be 5 times 2 squared minus 20 times 2 plus 15 which is equal to let's see this is 5 times 4 which is 20 minus 40 which is negative 20 plus 15 is negative 5 so just like that we're able to figure out the coordinate this coordinate right over here is the point 2 comma negative 5 now it's not so satisfying just a plug and chug a formula like this and we'll see where this comes from when you look at the quadratic formula this is kind of the first term it's that it's the x-value that's halfway in between the roots so that's one way to think about it but another way to do it and this is probably will be a more lasting help for you in your life because you might forget this formula and so really just try to Rima nip you late this equation so you can spot its minimum point and we're going to do that by completing the square so let me rewrite that so we have y is equal to and I'll write it as and what I'll do is out of these first two terms I'll factor out a 5 because I'm going to complete a square here and I'm going to leave this 15 out to the right because I'm gonna have to manipulate that as well so all right that is 5 times x squared minus 4x and then I have this 15 out here 15 out here and I want to write this as a perfect square and we just have to remind ourselves that if I have X plus a squared that's going to be x squared plus 2ax plus a squared so if I want to turn something that looks like this if I want to turn something like this to a X into a perfect square I just have to take half of this coefficient and square it and add it right over here in order to make it look like that so I'm going to do that right over here so if I take 1/2 of negative 4 that's negative 2 if I square it that is going to be positive 4 I have to be very careful here I can't just willy-nilly add a positive 4 here I have an equality here if they were equal before adding the 4 they're not going to be equal after adding the 4 so I have to do proper accounting here I either have to add 4 to both sides or actually be careful I have to add the same amount to both sides or subtract the same amount again now the reason why I was careful there is I didn't just add 4 to the right-hand side of the equation remember the 4 is getting multiplied by 5 I have added 20 to the right-hand side of the equation so if I want to make this balanced out if I want the Equality to still be true I either have to now add 20 to Y or have to subtract 20 from the right-hand side so I'll do that I'll subtract 20 from the right-hand side so I added five times four if you were to distribute this you'll see that I could have literally up here said hey I'm adding 20 I'm adding 20 and I'm subtracting 20 this is the exact same thing that I did over here if you distribute the 5 it becomes 5x squared minus 20x plus 20 plus 15 minus 20 exactly what's up here the whole point of this is that now I can write this in an interesting way I could write this as y is equal to 5 times X minus 2 squared and then 15 minus 20 is minus 5 so the whole point of this is now to be able to inspect this when does this equation hit a minimum value well we know that this we know that this term right over here is always going to be non-negative it's always going to be non-negative or we could say it's always going to be greater than or equal to zero so it's going this whole thing is going to hit a minimum value when this term is equal to zero or when x equals two when x equals two we're going to hit a minimum value and when x equals two what happens well this whole term is zero and Y is equal to negative five the vertex is to negative 5