Easy Solution for x + √x = 90: List of Solutions | Midterm Math Troubles

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The equation x + √x = 90 requires listing all solutions. The user initially identified x = 81 as a solution but struggled to derive it algebraically. They attempted to isolate the radical and square both sides, leading to incorrect results. The breakthrough came when they realized they needed to square the entire expression (90 - x) instead of just the radical. This correction clarified the path to solving the equation accurately.
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Homework Statement



Okay, so I just got home a few moments ago from the first of two midterms that I will have to take this year, (do the math on that one) and I can not seem to figure out the very first problem that was listed on the test.

It was

x + \sqrt{x} = 90


Homework Equations



It says "List all solutions"

The Attempt at a Solution



I tried several things, but none that seemed to fit correctly. Just by looking at it I can see that one of the solutions is x=81, but I can't get it to spit that out algebraically.

I've isolated the radical, then squared both sides to give myself a quadratic, but when I solve that I get two answers that do not work.

I'm sure this is an easy question, and one that I should know. It will probably wake me up in the middle of the night, but it's driving me crazy for now.
 
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Oh good grief... I got it.

I wasn't squaring 90-x together and foiling.

x = (90-x)^2

I was squaring them individually

x = 8100+x^2
 

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