SOLUTION: Hi. I'm an adult musician who sometimes uses mathy stuff to generate musical structures. I took algebra a long, long time ago and have forgotten most of it. A few years ago I stumb

Algebra ->  Equations -> SOLUTION: Hi. I'm an adult musician who sometimes uses mathy stuff to generate musical structures. I took algebra a long, long time ago and have forgotten most of it. A few years ago I stumb      Log On


   



Question 149055: Hi. I'm an adult musician who sometimes uses mathy stuff to generate musical structures. I took algebra a long, long time ago and have forgotten most of it. A few years ago I stumbled upon this idea: for any odd number, the sum of the integers below that number is divisible by the odd number. For example, if I take 3, the sum of the integers below 3 -- 1 + 2 = 3, which is divisible by 3. Or 5: 1+2+3+4 = 10, which is divisible by 5. I got a friend to help me with an equation that would help me figure out the sum, so if I choose a number like 83, I don't have to spend 5 minutes punching 1+2+3 and so on to figure out the sum. That equation was x(x-1)/2 = the sum of the integers below x, when x is an odd number.
Now, for the first time I'm trying to run this backwards and I can't remember how to solve for it. In other words, let's call "the sum of the integers below x," y. I know what y is, but I can't figure out x. For example, I've got 368 beats that I want to group in these different cycles. So I know that x(x-1)/2 = 368. I cross multiply the denominator and I get x(x-1) = 736. Then I multiply the x through and I get x^2 - x = 736. And then I'm stuck and don't know what to do to solve for x. I'm sure I'm doing something wrong or have forgotten some important step, because it's been forever since I've had to think about this stuff. Your help would be greatly appreciated.
cheers,
Andrea

Found 2 solutions by mangopeeler07, jim_thompson5910:
Answer by mangopeeler07(462) About Me  (Show Source):
You can put this solution on YOUR website!
x(x-1) = 736

x^2 - x = 736

x^2-x-736=0

(x-)(x+)???

This is because there are no two consecutive numbers who multiplied by each other give you 736. The closest there is to that is (26 and 28.307692) and (27 and 27.259259).

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

x%5E2-x=736 Start with the given equation.


x%5E2-x-736=0 Get all terms to the left side.


Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc where a=1, b=-1, and c=-736


Let's use the quadratic formula to solve for x


x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


x+=+%28-%28-1%29+%2B-+sqrt%28+%28-1%29%5E2-4%281%29%28-736%29+%29%29%2F%282%281%29%29 Plug in a=1, b=-1, and c=-736


x+=+%281+%2B-+sqrt%28+%28-1%29%5E2-4%281%29%28-736%29+%29%29%2F%282%281%29%29 Negate -1 to get 1.


x+=+%281+%2B-+sqrt%28+1-4%281%29%28-736%29+%29%29%2F%282%281%29%29 Square -1 to get 1.


x+=+%281+%2B-+sqrt%28+1--2944+%29%29%2F%282%281%29%29 Multiply 4%281%29%28-736%29 to get -2944


x+=+%281+%2B-+sqrt%28+1%2B2944+%29%29%2F%282%281%29%29 Rewrite sqrt%281--2944%29 as sqrt%281%2B2944%29


x+=+%281+%2B-+sqrt%28+2945+%29%29%2F%282%281%29%29 Add 1 to 2944 to get 2945


x+=+%281+%2B-+sqrt%28+2945+%29%29%2F%282%29 Multiply 2 and 1 to get 2.


x+=+%281%2Bsqrt%282945%29%29%2F%282%29 or x+=+%281-sqrt%282945%29%29%2F%282%29 Break up the expression.


So our answers are x+=+%281%2Bsqrt%282945%29%29%2F%282%29 or x+=+%281-sqrt%282945%29%29%2F%282%29


which approximate to x=27.634 or x=-26.634


I'm not sure what you're looking for, but if you aren't expecting to get an irrational answer, then try y=378