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CCSS.Math: , , ,

we're told the tailor opened a restaurant the net value of the restaurant in thousands of dollars t months after its opening is modeled by v of t is equal to 2t squared minus 20 t taylor wants to know what the restaurants lowest net value will be let me underline that and when it will reach that value so let's break it down step by step the function which describes how the value of the restaurant the net value of the restaurant changes over time is right over here if I were to graph it I can see that the coefficient on the quadratic term is positive so it's going to be some form of upward-opening parabola I don't know exactly what it looks like we could think about that in a second and so it's going to have some point right over here which really is the vertex of this parabola where it's going to hit its lowest net value and that's going to happen at some time T if you could imagine that this right over here is the T axis so my first question is is there some form is there some way that I can rewrite this function algebraically so it becomes very easy to pick out this low point which is essentially the vertex of this parabola pause this video and think about that all right so you could imagine the form that I'm talking about is vertex form where you can clearly spot the vertex and the way we can do that is actually by completing the square so the first thing I will do is actually let me factor out a 2 here because 2 is a common factor of both of these terms so V of T would be equal to 2 times T squared minus 10 T and I'm going to leave some space because completing the square which gets us to vertex form is all about adding and subtracting the same value on one side so we're not actually changing the value of that side but writing it in a way that so we have a perfect square expression and then we're probably going to add or subtract some value out here now how do we make this a perfect square expression and if all if any of this business about completing the square looks unfamiliar to you I encourage you to look up completing the square on Khan Academy and review that but the way that we complete the square is we look at this first degree coefficient right over here it's negative ten and we say all right well let's take half of that and square it so half of negative 10 is negative 5 and if we were to square it that's 25 so if we add 25 right over here then this is going to become a perfect square expression and you can see that it would be equivalent to this entire thing if we add 25 like that is going to be equivalent to t minus 5 squared just this part right over here that's why we took half of this and we squared it but as I alluded to a few seconds ago or a few minutes ago you can't just willy-nilly add 25 to one side or to one side of an equation like this that will not that will make this equality no longer true and sin fact we didn't just add 25 remember we have this 2 out here we added 2 times 25 you can verify that if you redistribute the 2 you'd get 2t squared minus 20 T plus 50 plus 2 times 25 so in order to make the quality or commit in order to allow it to continue to be true we have to subtract 50 so just to be clear this isn't some kind of you know strange thing I'm doing all I did is add 50 and subtract 50 you're saying wait you added 25 not 50 no look when I added 25 year it's in a parentheses and that whole expression is multiplied by 2 so I really did mult add 50 here so then I subtract 50 here to get to what I originally had and when you view it that way now V of T is he going to be equal to 2 times this business which we already established as t minus 5 squared and then we have the minus 50 now why is this form useful this is vertex form it's very easy to pick out the vertex it's very easy to pick out when the low point is the low point here happens when this part is minimized and this part is minimizing but you have to times something squared so if you have something squared it's going to hit its lowest point when this something is 0 otherwise it's going to be a positive value and so so this part right over here is going to be equal to zero when T is equal to five so the lowest value is when T is equal to five let me do that in a different color I don't want to reuse the colors too much so if we say V of five is going to be equal to two times five minus five trying to keep up with the colors minus five squared minus 50 notice this whole thing becomes zero right over here so V of five is equal to negative 50 that is when we hit our low point in terms of the net value of the restaurant so T represents months so we hit our low point we wrote our R we rewrote our function in a forum in vertex form so it's easy to pick out this value and we see that this low point happens at T equals five which is at time five months and then what is that lowest net value well it's negative 50 and remember the function gives us the net value in thousands of dollars so it's Nate it's a negative $50,000 is the lowest net value of the restaurant and you might say how do you have a negative value of something well imagine if say the building is worth $50,000 but you the the restaurant oh is a hundred thousand dollars that it would have a negative $50,000 net value