Understanding Logic Behind x=0: Explained

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In summary, the equation x(x-1)=0 can be solved by setting each factor equal to 0, resulting in two solutions: x=0 and x=1. This is because if the product of two numbers is 0, then one or the other of the numbers must be 0. The zero factor property states that if ab=0, then either a=0 or b=0, which holds true for the real and complex numbers but not for other number systems such as the integers modulo 6.
  • #1
emergentecon
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Homework Statement



Solve for x

x(x-1)=0

Homework Equations





The Attempt at a Solution



x = 0 and x = 1

what I am trying to understand is the logic behind the x=0?
could someone please explain that to me?
 
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  • #2
How much is zero times ANYTHING ?
 
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  • #3
phinds said:
How much is zero times ANYTHING ?

Zero
 
  • #4
Are you dividing by x and x-1 to get the answer?
 
  • #5
emergentecon said:

Homework Statement



Solve for x

x(x-1)=0

[offtopic]
This is ##x^2-x=0##
hmm? :confused:
So ##x^2=x##
How is this possible?
[/offtopic]
 
  • #6
adjacent said:
[offtopic]
This is ##x^2-x=0##
hmm? :confused:
So ##x^2=x##
How is this possible?
[/offtopic]

Well if x = 0 and x = 1 then it holds?
 
  • #7
emergentecon said:

Are you dividing by x and x-1 to get the answer?


##(x)(x-1)=0##
As you do with Quadratic equations,
##x=0##
##x-1=0##
so ##x=1##.That's it.Right?phinds
 
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  • #8
you have a number on the left, and a number on the right...
x(x-1) is a number, so is 0... and you want them to be equal...
when can x(x-1) be equal to 0?
you have two possibilities...
either x=0, so you will have 0*(0-1)=0*(-1)=0
or x=1, so you will have 1*(1-1)=1*0=0
so in both these cases you achieved what the equation asked for you x(x-1)=0
you didn't divide,multiply or anything...
 
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  • #9
ChrisVer said:
you have a number on the left, and a number on the right...
x(x-1) is a number, so is 0... and you want them to be equal...
when can x(x-1) be equal to 0?
you have two possibilities...
either x=0, so you will have 0*(0-1)=0*(-1)=0
or x=1, so you will have 1*(1-1)=1*0=0
so in both these cases you achieved what the equation asked for you x(x-1)=0
you didn't divide,multiply or anything...

You don't have to guess things.That's done in the way I mentioned above.
 
  • #10
it's not guessing... it's in fact what happens with factorizing anything (eg a polynomial equation)
 
  • #11
The obvious fact that if x= 0 then x(x- 1)= 0 and that if x= 1 then x-1= 0 so x(x- 1)= 0 shows that x= 0 and x= 1 are solutions but does NOT show that they are the only solutions. For example in the "integers modulo 6" it is true that 0 times anything is 0 so that x= 0 is a solution to 3x= 0 but so is x= 2.

For that you need the "zero factor property" some times phrased as "the set of real numbers (or complex numbers) does not have "zero divisors":

If ab= 0 then either a= 0 or b= 0 Which is not true for the "integers modulo 6".
 
  • #12
adjacent said:
This is ##x^2-x=0##
hmm? :confused:
So ##x^2=x##
How is this possible?
It seems like you're going backwards here, going from x2 - x = 0 to x2 = x.
The OP already had the left side of the equation in factored form (i.e., x(x - 1) = 0). Expanding the left side and adding x to both sides doesn't buy you anything. The important principle here is that if the product of two numbers is zero, then one or the other of the numbers has to be zero.
 
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  • #13
Mark44 said:
It seems like you're going backwards here, going from x2 - x = 0 to x2 = x.
The OP already had the left side of the equation in factored form (i.e., x(x - 1) = 0). Expanding the left side and adding x to both sides doesn't buy you anything. The important principle here is that if the product of two numbers is zero, then one or the other of the numbers has to be zero.

ah,Ok I've got it.Thank you. :smile:
 

Related to Understanding Logic Behind x=0: Explained

1. What is the significance of x=0 in logic?

In logic, x=0 represents a statement or equation where the value of x is equal to zero. This can be used to determine the truth value of a logical statement or to solve equations.

2. How is x=0 related to mathematical equations?

In mathematical equations, x=0 represents a solution where the value of x is equal to zero. This can be used to solve equations and find the root or point of intersection.

3. Can x=0 have different meanings in different contexts?

Yes, the meaning of x=0 can vary depending on the context in which it is used. In logic, it represents a truth value or solution to an equation, while in other fields it may have a different interpretation.

4. How does understanding the logic behind x=0 help in problem-solving?

Understanding the logic behind x=0 can help in problem-solving by providing a clear understanding of how to approach and solve equations. It also helps in determining the truth value of logical statements.

5. Are there any real-world applications of understanding the logic behind x=0?

Yes, there are many real-world applications of understanding the logic behind x=0, such as in engineering, computer science, and economics. It can also be used in everyday problem-solving and decision making.

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