Can x be written as both a positive and negative value in an equation?

In summary, The use of ##\pm \sqrt{}## in formulas allows for both positive and negative solutions to be considered. However, when taking the square root of a number, only the positive solution is returned. Also, when solving equations, it is important to consider restrictions on the variable in order to determine which solution is appropriate.
  • #1
LiHJ
43
2
<< Mentor Note -- OP has been advised to type their questions into the forum next time, instead of inserting images >>

1. Homework Statement

Dear Mentors and PF helpers,

I have a question from today's lesson.

image.jpg


Homework Equations

The Attempt at a Solution



Is 1) and 2) both acceptable? If only of of them is acceptable, is there any reason why?

Thanks for your time
 
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  • #2
Only 2.
The ## \sqrt{} ## implies that it only returns the positive square root of a number. That is why you will often see ##\pm \sqrt{} ## in formulas.
 
  • #3
For x any real number, [itex]\sqrt{x^2}= |x|[/itex], NOT [itex]\sqrt{x}= x[/itex]

That is easily seen by taking x to be any negative number, say x= -2. In that case, [itex]x^2= (-2)^2= 4[/itex] and [itex]\sqrt{x^2}= \sqrt{4}= 2[/itex], NOT -2.
 
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  • #4
However, if you know something about x, like it is positive, then you may use the notation in (1) .
Also, if you have ##y = x^2##, then ##x = \pm \sqrt{y}##.
 

Related to Can x be written as both a positive and negative value in an equation?

1. What is algebra?

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols to solve equations and represent relationships between quantities.

2. What are variables?

Variables are symbols or letters that represent unknown quantities in algebraic expressions or equations. They can take on different values, which allows us to solve for the unknown quantity.

3. What is an equation?

An equation is a mathematical statement that shows the equality of two expressions. It consists of an equal sign and two expressions on either side.

4. How do you solve for an unknown variable?

To solve for an unknown variable in an equation, you need to isolate the variable on one side of the equal sign by using inverse operations. This will allow you to find the value of the variable that makes the equation true.

5. Why is algebra important?

Algebra is important because it helps us solve real-world problems by representing them in mathematical form. It also provides the foundation for more advanced areas of mathematics and other subjects like physics, economics, and engineering.

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