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Math Help - Simplifying expressions

  1. #1
    Member purplec16's Avatar
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    Simplifying expressions

    \frac{x+2}{2x-3}-\frac{x^2-4}{2x^2-3x}

    The problem is not simplifying correctly, somethings supposed to cross out and I'm not getting it
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  2. #2
    Super Member Bacterius's Avatar
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    \frac{x+2}{2x-3}-\frac{x^2-4}{2x^2-3x}

    Put under common denominator :

    \frac{(2x^2-3x)(x+2)}{(2x-3)(2x^2-3x)}-\frac{(2x-3)(x^2-4)}{(2x-3)(2x^2-3x)}

    Expand all this stuff :

    \frac{(2x^2-3x)(x+2) - (2x-3)(x^2-4)}{4x^3 - 6x^2 - 6x^2 + 9x}

    Expand further :

    \frac{(2x^3 + 4x^2 - 3x^2 - 6x) - (2x^3 - 8x - 3x^2 + 12)}{4x^3 - 12x^2 + 9x}

    Again :

    \frac{2x^3 + 4x^2 - 3x^2 - 6x - 2x^3 + 8x + 3x^2 - 12}{4x^3 - 12x^2 + 9x}

    Cancel out terms :

    \frac{4x^2 + 2x - 12}{4x^3 - 12x^2 + 9x}

    Factorize both polynomials :

    \frac{(4x - 6)(x + 2)}{x(4x - 6)(x - \frac{ 3}{2})}

    Cancel out :

    \frac{x + 2}{x(x - \frac{ 3}{2})}

    Finally :

    \frac{x + 2}{x^2 - \frac{ 3}{2}x}

    I think it's right, I haven't checked though ...
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  3. #3
    Member purplec16's Avatar
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    Thanks so much I checked and I think it's write but I thought the common demonimator means that its suppose to be "2x^2-3x" for the whole equation
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  4. #4
    Super Member Bacterius's Avatar
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    When adding/substracting fractions, all fractions need to be under the same denominator, otherwise it won't work. To achieve this, we usually multiply both parts of fraction 1 by the denominator of fraction 2, and the other way round. Then, we substract (in your case) the fractions. (we are not substracting the denominators, obviously)

    EDIT : did you see that ? 77 posts, and 7 thanks ... it's my lucky day ...
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  5. #5
    Member purplec16's Avatar
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    So aren't you multiplying to get both common denominators in both fractions so that you can minus
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  6. #6
    Super Member Bacterius's Avatar
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    Quote Originally Posted by purplec16 View Post
    So aren't you multiplying to get both common denominators in both fractions so that you can minus
    That's what I said, yes. You multiply each member of one fraction by the denominator of the other, then you substract the top members of both fractions and put it all under the common denominator.
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  7. #7
    Member purplec16's Avatar
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    but I thought in the end when you do that your supposed to have the same denominator which is "2x^2-3x"
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  8. #8
    Super Member Bacterius's Avatar
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    Quote Originally Posted by purplec16 View Post
    but I thought in the end when you do that your supposed to have the same denominator which is "2x^2-3x"
    No, the common denominator would be both denominators multiplied together, that is :

    (2x - 3)(2x^2 - 3x)

    And when expanded, simplified, factorized and cancelled out by simplification, you find something else.
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  9. #9
    Member purplec16's Avatar
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    oh ok thank you I get it now

    nite
    Last edited by purplec16; November 9th 2009 at 07:34 PM. Reason: idk
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  10. #10
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    Quote Originally Posted by Bacterius View Post
    When adding/substracting fractions, all fractions need to be under the same denominator, otherwise it won't work. To achieve this, we usually multiply both parts of fraction 1 by the denominator of fraction 2, and the other way round. Then, we substract (in your case) the fractions. (we are not substracting the denominators, obviously)

    EDIT : did you see that ? 77 posts, and 7 thanks ... it's my lucky day ...
    No, we don't "usually multiply both parts of fraction 1 by the denominator of fraction 2, and the other way round." We usually multiply numerator and denominator of each fraction by whatever is necessary to ge the "least common denominator".

    In the original fractions, the denominators are 2x-3 and 2x^2- 3x= x(2x-3). The least common denominator is x(2x-3)= 2x^2- 3x. We need to multiply numerator and denominator of the first fraction, \frac{x+2}{2x-3} by x to get \frac{x^2+ 2x}{x(2x-3)}. Since the denominator of the second fraction is already x(2x-3), we don't have to change that.
    \frac{x+2}{2x-3}- \frac{x^2- 4}{2x^2- 3x}= \frac{x^2+ 2x}{x(2x-3)}- \frac{x^2-4}{x(2x-3)} \frac{x^2+ 2x- x^2+ 4}{x(2x-3)}= \frac{2x+ 4}{x(2x-3)}
    You can, of course, multiply that denominator out to get \frac{2x+4}{2x^2- 3x}.
    Last edited by mr fantastic; November 10th 2009 at 03:49 AM. Reason: Fixed some latex
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  11. #11
    Super Member Bacterius's Avatar
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    Ah yes, I didn't think of that.
    Surely because when I was in junior I used to multiply the two denominators (typically 7 and 9) together to be sure to get a common divisor, because I wouldn't want to think about it.
    I must think of doing this little check ... thanks HallsofIvy
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