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Hmm... interesting. I see that the second equation is just as nasty as the first one (the only difference is a t-1 vs. a t-2). But I bet the big picture idea is still the same. :) 1) Set equal 2) Solve for t 3) Check answer.
This One Is Tricky .
Yeah, I notice that on the surface we also have a weird (t-1)^2 and (t-2)^2 expansion needed too. We have to "expand the binomial" in math speak.
Let's try it and see what we get. We'll start with the first equation -5(t-1)^2 + 5
Alright, well I don't like the whole parenthesis squared thing so let's remember that exponents just mean to multiply the same thing twice: -5(t-1)^2 + 5 --> -5(t-1)(t-1) + 5
Does that make sense so far?
Alright, keep track of what you are thinking and what is confusing. Let me finish the full thought here and see if that helps. :)
We rewrite: -5(t-1)^2 + 5 -> -5(t-1)(t-1) + 5 then, we use FOIL or distribution (whichever you learned) to get: -5(t-1)(t-1) + 5 -> -5( t^2 - t - t + 1) + 5 simplifying: -5(t^2 - t - t + 1) + 5 -> -5(t^2 - 2t + 1) + 5 distribute the -5 (watch out for sign changes!): -5(t^2 - 2t + 1) + 5 -> -5t^2 + 10t - 5 + 5 simplifying: -5t^2 + 10t - 5 + 5 -> -5t^2 + 10t I know that is a lot, but see if you can follow the thought process. If you have any questions, I'll be happy to try and answer them. There are several algebra ideas floating around here.
Look Like Something I Seen Before
That's a good thing... I suppose. When it comes to math, we're basically saying "Yep, I've seen Godzilla before." :)
Lol Right .
Ahh! Error.... Here it is corrected. :) Well, if you are brave, we can do the same thing with the other equation -5(t-2)^2 + 5. If we do it carefully, we get -5(t-2)^2 + 5 -5(t-2)(t-2) + 5 -5(t^2 - 2t - 2t + 4) + 5 -5(t^2 - 4t + 4) + 5 -5t^2 + 20t - 20 + 5 -5t^2 + 20t - 15 Yikes... but, OK. :P Should have been -15 at the end.
ok got it
Is there any step in particular that is confusing? Hopefully most of the make sense. The concepts here (in case you remember the names, which I never did): expand binomial: (t-1)^2 = (t-1)(t-1) FOIL: (t-1)(t-1) = first + outer + inner + last = t*t + t*-1 + -1*t + -1*-1 = t^2 -t -t + 1 = t^2 - 2t + 1
Oh, well you said you got it, but just checking... :)
i was saying i got the 1st step lol but the whole binomial never heard of it
No worries, you may have heard it called something different. Terminology is replaceable. :)
i heard of terminology
Well, now we are actually through most the hard part of this question. 1) Expand the equations we got -5(t-1)^2 + 5 ---> -5t^2 + 10t and -5(t-2)^2 + 5 ---> -5t^2 + 20t - 15 All we do now is set these two equations equal to each other, and proceed as before.
soooo -5t^2 +10t=-5t^2 + 20t - 15
Right! Now, we can choose to move one side or the other it doesn't matter. I'll move the left side since it is shorter (less steps). :)
-5t^2 + 10t = -5t^2 + 20t - 15 +5t^2 +5t^2 ------------------------ 10t = 20t - 15 ooooh nice, the t^2's are gone! 10t = 20t - 15 -10t -10t ------------- 0 = 10t - 15 I don't like it written backwards, so I will spin it around 10t - 15 = 0 There. We don't even need the quadratic formula for this one!
yay this one was fun but different
Yes, the mathematical definition of "fun" is "utterly tortuous and ridiculous". :) If you get into math at the master's level the definition of an "interesting problem" is one that "you have no clue how to solve". lol
Just to finish it off (though I'm sure you already finished the missing steps)... 10t = 15, so t = 15/10 = 1.5.
I've got to head out now. But I do want to give you a link to a very good resource: Check out desmos.com It is a very powerful graphing calculator that lets you "see" the answer visually. It is not a replacement for knowing the algebra (in class you still need to solve it manually), but I think it is cool to know that there is technology that does these problems instantly. :) For example, I did this problem we just solved: https://www.desmos.com/calculator/1h08m4ibe7
thanks , ill stay in touch your awesome
Of course, whoever wrote the code for the computer needed to know the handwritten algebra first... :) Thanks! I hope you got something out of all the crazy math notation I threw at you. It's still weird that little symbols allow us to communicate ideas... crazy stuff... Take care! :)