The Quiz Begat a Quiz

To get the payout amounts for the fantasy football league, I started with two numbers: the $360 total that was in the prize kitty, and the $30 payout for the third place winner, which is exactly the amount of the initial pay-in. With these two numbers, I attemped to solve the following equation:

$30 + $30x + $30x-square = $360,

with the second ($30x) and third ($30x-square) amounts being the second and first place payouts.

I figured out x needed to equal about 2.85 for the equation to work. But I solved it by trial and error.

I'm wondering what is the actual mathematical method to solve the equation above.

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