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I have holes in my math, help me.

  1. Sep 27, 2013 #1
    What is happening here?
    x2 – x – 2 = 0
    (x – 2)(x + 1) = 0
    x = 2 or x = –1


    This is related to domain and ranges, finding the possibilities of the domain.



    I would like to just see the process of how to solve this and then knowing what this actually is in mathematical terms (I'm assuming it is basic algebra, but what within algebra is this considered?)

    Sorry for my blatant ignorance, I am getting back into math right now.
     
  2. jcsd
  3. Sep 27, 2013 #2

    mfb

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    2016 Award

    Staff: Mentor

    x2 means x2?
    Going from the first to the second line is a bit tricky (basically clever guessing), but can you derive the first line from the second?

    To come from the second to the third: how can a product of two factors be zero? There are two options, and those give you the solutions.
     
  4. Sep 27, 2013 #3

    Mark44

    Staff: Mentor

    I'm guessing you mean x2 - x - 2 = 0. I used HTML tags to make the superscript exponent. A simpler way is to write an exponent using the ^ character, like this:
    x^2 -x - 2 = 0
    The equation you wrote is a quadratic equation, so called because the highest degree term (x2) is a squared term (quadratus is Latin for square).

    What was done in the work you show is called factoring, with the idea that if you know that two things multiply to make zero, then one or the other of them must be zero.
     
  5. Sep 28, 2013 #4

    verty

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    If you need book recommendations, ask in the book section and you'll certainly get help.

    The process used in this question is called factoring a quadratic equation, it is a shorter method than the longer "completing the square", although there is also the quadratic formula that one can use.
     
  6. Sep 28, 2013 #5

    HallsofIvy

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    Oh, dear. I wonder if you are not spending too much time memorizing formulas and not enough learning what those formulas mean. Because here you see to be spending a lot of time trying to do something that is not related to solving the given problem! You say this has to do with "finding the possibilities of the domain". If that is the case, it has nothing at all to do with factoring or finding what values of x makes the function value 0.

    To find the "natural domain" of a function given by a formula, you need to determine any value of x for which the operations cannot be performed. (For example division by 0 cannot be performed and you cannot take the square root of a negative number.) Here, the only operations are the basic arithmetic operations of addition and multiplication- and you can do those to any number. The domain of this function is "the set of all real numbers.

    Nor do factoring and solving an equation have anything to do with finding the range. Since this function is quadratic, completing the square can be useful. f(x)= x^2- x- 2= (x^2- x+ 1/4)- 1/4- 2= (x- 1/2)^2- 9/4. (I have used the fact that any "square" is of the form (x- a)^2= x^2- 2ax+ a^2. That is the constant term must be half the coefficient of x, squared. Here, the coefficient of x is -1. Half of that is -1/2 and, squaring, 1/4. I added and subtracted 1/4, so as not to change the actual value, getting x^2- x+ 1/4- 1/4- 2, then wrote x^2- x+ 1/4 as x^2- 2(1/2)x+ (1/2)^2= (x- 1/2)^2.

    The point is that a square is never negative (x- 1/2)^2 can be 0 (for x= 1/2) and can be any other positive number but never 0. So (x- 1/2)^2- 9/4 can be equal to -9/4 and can be any number larger than -9/4 but can never be less than -9/4. The range is "the set of all numbers larger than or equal to -9/4."
     
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