Solving Square Root & Quadratic Equations

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Discussion Overview

The discussion revolves around the manipulation of square root equations and the implications of rearranging terms within those equations. Participants explore whether certain algebraic transformations are valid and how they apply to both square root and quadratic equations.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant proposes starting with the equation 0 = √x - √x and questions if it can be rearranged to 0 = 2√x.
  • Another participant challenges the assumption that a - a = 2a, questioning the validity of this transformation.
  • Some participants assert that if something equals 0, the signs can be flipped, suggesting that -x = 0 if x = 0.
  • There is a discussion about the nature of the solutions to the equation 0 = a - a, with some suggesting it is undefined for any number.
  • One participant clarifies that the equation 0 = √x - √x is true for any nonnegative real number x, emphasizing that √x has one value when x ≥ 0.
  • Another participant points out that the transformation from √x - √x = 0 to √x + √x = 0 is not valid, indicating that the two equations are not equivalent.
  • There is confusion expressed by some participants regarding the original question and the intent behind the transformations.

Areas of Agreement / Disagreement

Participants express differing views on the validity of algebraic manipulations involving square roots. There is no consensus on whether the proposed transformations are correct or meaningful, and the discussion remains unresolved.

Contextual Notes

Participants highlight limitations in understanding the implications of manipulating square root equations and the conditions under which these manipulations hold true. There is also a lack of clarity regarding the original intent of the question posed.

Einstein's Cat
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Let's say there's an equation

0 = √x - √x

I intend to make x the subject of the equation; however because it is a square root, there are numerous solutions; however can I just assume that

0= √x - -√x= 2√x

Can I now just rearrange this equation to make x the subject? In other words is the equation above equivalent to the equation below?

0= -√x -√x = -2√x

Would the same be true if there were roots in a quadratic equation?
 
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Do you really expect that a-a=2a in general? Here a=sqrt(x).

What do you know about the solutions to the equation 0=a-a?
 
If something equals 0, you can flip the sign any way you want.
If x = 0, then -x = 0 as well
 
mfb said:
Do you really expect that a-a=2a in general? Here a=sqrt(x).

What do you know about the solutions to the equation 0=a-a?

I think that as this applies to any number; the solutions is any number and therefore the equation is undefined.
 
Einstein's Cat said:
Let's say there's an equation

0 = √x - √x
Do you realize that the right side is equal to zero for any nonnegative real number x?
Einstein's Cat said:
I intend to make x the subject of the equation; however because it is a square root, there are numerous solutions
Not sure what you mean by this. The symbol ##\sqrt{x}## has one value, assuming that ##x \ge 0##.
Einstein's Cat said:
; however can I just assume that

0= √x - -√x= 2√x
?
This is different from the equation you have at the top of your post.
The first equation you show is √x - √x = 0. The equation just above, when simplified is √x + √x = 0. These two equations are not equivalent.
Einstein's Cat said:
Can I now just rearrange this equation to make x the subject? In other words is the equation above equivalent to the equation below?

0= -√x -√x = -2√x

Would the same be true if there were roots in a quadratic equation?
I don't understand what you're asking here.
 
Mark44 said:
Do you realize that the right side is equal to zero for any nonnegative real number x?
Not sure what you mean by this. The symbol ##\sqrt{x}## has one value, assuming that ##x \ge 0##.
?
This is different from the equation you have at the top of your post.
The first equation you show is √x - √x = 0. The equation just above, when simplified is √x + √x = 0. These two equations are not equivalent.

I don't understand what you're asking here.
I apologise for I am unable to express what I mean; this thread serves no purpose
 
Einstein's Cat said:
I apologise for I am unable to express what I mean; this thread serves no purpose
Thread closed.
 

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