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Math Help - solving for A and B

  1. #1
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    solving for A and B

    1/((u^2)-4) = A/(u-2) + B/(u+2)

    how do u solve for A and B?

    i am told that A and B can be easily solved to be A=.25 B=-.25 but i dont see how to get that. Please help
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by imthatgirl View Post
    1/((u^2)-4) = A/(u-2) + B/(u+2)

    how do u solve for A and B?

    i am told that A and B can be easily solved to be A=.25 B=-.25 but i dont see how to get that. Please help
    Ahh, I'm going to assume that this is partial fractions decomposition? If so you can do this. Multiplying through by u^2-4 gives

    1=A(u+2)+B(u-2)\quad{\color{red}\star}

    Now we use this little trick, if we let u=2 we get

    1=4A+0B\implies{A=\frac{1}{4}}

    And letting u=-2 gives 1=0A-4B\implies{B=\frac{-1}{4}}

    The other way is to compare coefficents. Expanding the starred step gives

    1=\left(A+B\right)u+2A-2B

    Now we know that for this expression to be true that each of the coefficients must match. So A+B=0 and 2A-2B=1. Solving those gives the correct values.
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  3. #3
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    how do u know to let u=2 and -2?
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  4. #4
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by imthatgirl View Post
    how do u know to let u=2 and -2?
    Because think about it if you have something like

    C=A(x+d)+B(x+e)

    And you want to isolate either A or B you want the other one to dissapear, and how do you do that? You make its coefficient zero, so you basically need to use the value of x that makes either A or B zero. In this case it would be x=-d to Isolate B and x=-e for A
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  5. #5
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    Quote Originally Posted by imthatgirl View Post
    1/((u^2)-4) = A/(u-2) + B/(u+2)

    how do u solve for A and B?

    i am told that A and B can be easily solved to be A=.25 B=-.25 but i dont see how to get that. Please help
    Consider:

    \frac{A}{u-2}+\frac{B}{u+2}=\frac{A(u+2)+B(u-2)}{(u-2)(u+2)}=\frac{u(A+B)+2(A-B)}{u^2-4}

    Now if:

    \frac{A}{u-2}+\frac{B}{u+2}=\frac{u(A+B)+2(A-B)}{u^2-4}=\frac{1}{u^2-4}

    we must have:

    u(A+B)+2(A-B)=1

    so:

    A+B=0

    and

    A-B=\frac{1}{2}

    Solve these and you will get the given answers.

    CB
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