Solving for x with a Square Root Sign: Help Needed!

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Discussion Overview

The discussion revolves around solving the equation x + √x = 6, focusing on methods for isolating the variable x in the presence of a square root. Participants explore various approaches and techniques relevant to algebraic manipulation and inverse functions.

Discussion Character

  • Homework-related, Mathematical reasoning, Technical explanation

Main Points Raised

  • One participant expresses confusion about solving for x due to the square root and mentions that squaring both sides initially complicated the problem.
  • Another participant suggests isolating the square root to simplify the equation.
  • A different approach involves substituting u = √x to transform the equation.
  • One participant notes that the equation can be treated as a quadratic in √x and proposes factorization as a method to solve it.
  • Another participant recommends rewriting the equation to isolate √x and cautions that squaring both sides can introduce spurious roots, emphasizing the need to verify solutions in the original equation.

Areas of Agreement / Disagreement

Participants present multiple methods for solving the equation, indicating a lack of consensus on a single preferred approach. Various techniques are proposed, but no agreement on the best method is reached.

Contextual Notes

Some participants highlight the potential complications introduced by squaring both sides of the equation, suggesting that care must be taken to check solutions against the original equation.

Jules18
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Does anyone know how to solve for x in the following equation:

x + [tex]\sqrt{x}[/tex] = 6

I don't know how to solve for x with eq'ns like this, and I'm studying inverse fxns right now, so I'm told that's what I'm supposed to do.

The square root sign is throwing me off.
The first time I tried, I tried squaring both sides to get rid of the sq. root sign, but it just made it more complicated.
Any suggestions?

~Jules~
 
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you need to isolate the sqrt(x), it will make everything A LOT nicer looking
 
Try substituting u = sqrt(x).
 
It is a quadradic in sqrt(x)
factorize into the form
x+sqrt(x)-6=(a*sqrt(x)+b)(c*sqrt(x)+d)
 
Three basically different methods, all of which work! I would prefer emyt's method:
write the equation as [itex]\sqrt{x}= 6- x[/itex] and square both sides. Caution: squaring both sides of an equation (or, more generally, multiplying both sides of an equation by something involving the unknown) can introduce "spurious roots" so be sure to check any solution in the original equation.
 

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