By the way, did you notice something about this week's POTW posting that is different from the rest? If you know what that is, tell me and I ensure you there is a reason for it!
Challenge!
Monday, June 19, 2017
Final POTW of the Year! A Challenge!
As promised, here is the most challenging question of the year! It has been another great year for the POTW. Thanks to those who have contributed all year long and encouraged me to keep-up with adding the POTW with your excitement and interest in them! Special shout-out to Alan for his continuous interest in all-things-mathematics, especially the weekly POTW!
By the way, did you notice something about this week's POTW posting that is different from the rest? If you know what that is, tell me and I ensure you there is a reason for it!
Challenge!
By the way, did you notice something about this week's POTW posting that is different from the rest? If you know what that is, tell me and I ensure you there is a reason for it!
Challenge!
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can you please tell me if we use bedmas or not?
ReplyDeleteThis is quadratic
DeleteI haven't learned that yet so I just did it without bedmas. I still got the same thing as Alan though. Quadratic is probably faster so I'll try learning that.
DeleteI'm not sure if it went through or not, but did my comment for the answer go through or not because it was too long?
ReplyDeleteI'm going to re-type my previous comment just in case because I don't think it went. This time I'm going to shorten it. It took a while, but you can see the statement as y to the power of z = 1. y is (x to the power of 2 - 5 (x) + 5). z is x to the power of 2 + 4x - 60. I'm going to say it again since I don't think my previous comment went in. I didn't use bedmas since it'll take too long to wait for a response and I even tried doing bedmas and I couldn't do it. Repeating y to the power of z is 1. I know that -1 to the power of an even number is 1, 1 to the power of anything is 1 and anything to the power of 0 is 1. After a lot of trial and errors I discovered that the final answer of x are 1, 2, 4, 6, and -10. To show an example I'll use -10. -10 to the power of 2 is 100 - 5 (-10) + 5 is not 1 or -1. Z therefore must be 0. -10 to the power of 2 is 100 + 4 (-10) - 60 is 0. Therefore 10 works.
ReplyDeleteMy other comment explains much more of what I did, but it took like an hour to write up and I don't know if it got sent or deleted. If sent, then I'll have 2 comments, if not then this will have to do. Also, my other comment also asked this. Mr Milette, how do we check our answers?
does anyone agree that this POTW is more time consuming than hard. I feel like other POTWs were harder
DeleteThat's why this question is harder, it takes more time to figure it out.
Delete1 is impossible for a power as of the -60
Delete1 is impossible for the power because of the minus 60
DeleteHow? I'll use 1. (1 to the power of 2 is 1 - 5 (1) + 5 is 1) to the power of anything is 1 because 1 to the power of anything is 1. 1 is already possible.
DeleteOk, so pre-note on this:
ReplyDeleteI was missing a bunch of formulas to figure out this question so my mom had to explain to me how to do it.
So here is how I did the VERY CHALLENGING POTW thingy:
First of all, there are three cases.
1. The bottom part (x^2-5x+5) = 1.
2. The top part (x^2+4x-60)=0
3. The bottom part = -1 and the top part/the exponent is some even number.
Reasons:
If the bottom part is equal to one, no matter what the exponent is the amount will always equal to one.
2. If the exponent is 0, all numbers will turn out to be 1 EXCEPT IF THE BOTTOM NUMBER EQUALS TO 0.
3. If the bottom number is negative 1 and the exponent is a multiple of 2, -1*-1 = 1 so that is the answer we want.
Solving:
1. x^2-5x+5 = 1
x^2-5x+4 = 0
(x-1)(x-4) = 0
That sentence was made using the distributive law property.
This means that x can either be 1 or 4, since one of these two values has to equal to 0. If it was 1, then 1-1 would be 0, and if it was 4, 4-4 = 0.
So that is two different answers done!
Case 2:
x^2+4x-60 = 0
(x+10)(x-6) = 0
This means that x can equal to -10 or 6.
THAT IS 4 POSSIBILITIES DONE!!!!
Case 3:
x^2-5x+5 = -1
x^2-5x+6 = 0
(x-2)(x-3)=0
This gives us 2 possibilities: 2 or 3. However, only one of them work.
So, I plug in the two values and for the exponent it gives me -48 and -39, respectively. Using this, whichever one has an even value is the answer.
1/(-1)^39 = -1 and 1/(-1)48 = 1.
This means that the value I need is 48 because it matches up all of the -1s perfectly together, which means that the exponent is even.
x can also be equal to 2.
So overall, x can be equal to 1, 4, -10, 6, and 2.
5 answers!
And again, just saying, I take no credit for this, I figured out a few parts but without the formulas none of this would have happened.
Also, if anyone would like to try to solve it I know how to use the formulas so I can be some help(Even though I might get too confusing at times :D)
-Alan
"2. If the exponent is 0, all numbers will turn out to be 1 EXCEPT IF THE BOTTOM NUMBER EQUALS TO 0. "
Delete0 to the power of 0 IS 1
"Even though I might get too confusing at times"
Deleteyou're not that consuming and alan, at least you don't take a long time understanding what the question is really asking. At first I thought it meant something like the first x is a number, the second x is another number and so on. I didn't realize until after that there are 5 POSSIBLE answers
No, 0 to the power of 0 may be 1 on a calculator, but I'm pretty sure 0 to the power of 0 is actually undefined.
DeleteI read how it can be 0 (probably not because again anything divided by 0 is 1), 1, and undefined. Mathematicians say it's 1 sometimes or undefined.
Deleteconsider the equals 1 on the oppposite side
DeleteThe values of x that satisfy the equation are
Deletex = -10, x = 1, x = 2, x = 4 and x = 6. Way to go Alan. It was a high school question for a challenge at the end of the year. Don't fret if it was confusing or too difficult, as it is way beyond our current curriculum levels.
Is the different thing that there is no solution for the previous POTW?
ReplyDeleteThe values of x that satisfy the equation are
ReplyDeletex = -10, x = 1, x = 2, x = 4 and x = 6.