Can X=2 and X=-2 at the same time?

  • Context: High School 
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Discussion Overview

The discussion revolves around the mathematical implications of the equation X = the square root of 1, specifically addressing whether X can simultaneously equal both 1 and -1. Participants explore the logical and mathematical reasoning behind the square root operation and the nature of equations that yield multiple solutions.

Discussion Character

  • Debate/contested
  • Mathematical reasoning
  • Conceptual clarification

Main Points Raised

  • Some participants argue that if X = the square root of 1, then X can be either 1 or -1, leading to the claim that 1 = -1, which is challenged by others.
  • Others clarify that the square root of 1 is defined as the principal root, which is 1, and that the equation X^2 = 1 has two distinct solutions (1 and -1) but does not imply that X can take both values simultaneously in the same context.
  • Some participants express confusion over the concept of overlapping solutions and the implications of using X in equations without choosing a single value.
  • There are discussions about the nature of equations and whether they can yield multiple answers, with some asserting that an equation can have multiple solutions while others emphasize the need to select one value for X when substituting in equations.
  • Participants also discuss the distinction between using the square root function and the broader implications of equations that can have multiple solutions, such as cubic or quartic equations.

Areas of Agreement / Disagreement

Participants generally disagree on the interpretation of the square root and the implications of having multiple solutions. While some acknowledge that X can be either 1 or -1, they maintain that X cannot be both at the same time in a single equation. The discussion remains unresolved regarding the understanding of these concepts.

Contextual Notes

There are limitations in the discussion regarding the definitions of square roots and the assumptions about the nature of equations. The participants do not reach a consensus on the interpretation of overlapping solutions and the implications of choosing values for X.

Who May Find This Useful

This discussion may be of interest to those studying algebra, particularly the properties of square roots and the nature of equations with multiple solutions.

  • #31


ahh but root uses a double o and all the letters are on the same line ... its just more efficient haha
 
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  • #32


It may be more efficient but it's also ambiguous. We could use "root" to represent any nth root whereas "sqrt" specifically denotes the squareroot.
 
  • #33


also true
 
  • #34


so (sqrt 1 - sqrt 1) is different to
x=sqrt 1
x-x
 
  • #35


chalky00 said:
so (sqrt 1 - sqrt 1) is different to
x=sqrt 1
x-x
(sqrt 1 - sqrt 1) has 4 possible solutions.
 
  • #36


chalky00 said:
so (sqrt 1 - sqrt 1) is different to
x=sqrt 1
x-x

No! Hopefully without confusing you too much, if we let x = \sqrt{1} then x=1 because the \sqrt operation retrieves the principal (positive) square root. However, if x is a number such that x^2 = 1 then x=1 or x=-1. Now, using this second definition of x, once we choose a value for x we need to stick with it. We can't have x = 1 and x = -1 at the same time because this would violate the law of non-contradiction and you're already assuming that 1 = -1.
 
  • #37


Stop feeding the troll (OP)!
 

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