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

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) (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) (Show Source): You can put this solution on YOUR website!

Start with the given equation.

Get all terms to the left side.

Notice we have a quadratic equation in the form of where , , and

Let's use the quadratic formula to solve for x

Start with the quadratic formula

Plug in , , and

Negate to get .

Square to get .

Multiply to get

Rewrite as

Add to to get

Multiply and to get .

or Break up the expression.

So our answers are or

which approximate to or

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