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Math Help - Can anyone help me with algebra question

  1. #1
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    Can anyone help me with algebra question

    Hi,
    I am a mum who never learned algebra. My daughter is learning it at school and I am unable to help with homework. Is there someone out there kind enough to show me how to solve the below question so I can help my daughter out with similar problems.

    2W + 3W = W
    W+1 W-1


    Thanks
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  2. #2
    Super Member
    Joined
    May 2009
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    2W + 3W = W
    W+1 W-1

    \frac{2w}{w + 1} + \frac{3w}{w - 1} = w

    Getting rid of the fractions would be a first step. You do that by multiplying everything by a common denominator. The common denominator for w + 1 and w - 1 is simply their product, (w + 1)(w - 1):

    (w + 1)(w - 1)\left(\frac{2w}{w + 1} + \frac{3w}{w - 1}\right) = (w + 1)(w - 1)w

    What happens now? When you multiply (w + 1)(w - 1) by the first fraction the w + 1 cancels out, and you're left with 2w(w - 1). Likewise, when you multiply (w + 1)(w - 1) by the second fraction the w - 1 cancels out, and you're left with 3w(w + 1).

    2w(w - 1) + 3w(w + 1) = (w + 1)(w - 1)w

    As for the right side, you'll have to multiply it out. (w + 1)(w - 1) is a special product; it equals w^2 - 1. Then distribute the w over the w^2 - 1:

    2w(w - 1) + 3w(w + 1) = (w^2 - 1)w
    2w(w - 1) + 3w(w + 1) = w^3 - w

    Let's use the distributive property on the left side and start combining like terms:

    \begin{aligned}<br />
2w(w - 1) + 3w(w + 1) &= w^3 - w \\<br />
2w^2 - 2w + 3w^2 + 3w &= w^3 - w \\<br />
5w^2 + w &= w^3 - w \\<br />
0 &= w^3 - 5w^2 - 2w<br />
\end{aligned}

    Factor out a w:
    \begin{aligned}<br />
w^3 - 5w^2 - 2w &= 0 \\<br />
w(w^2 - 5w - 2) &= 0<br />
\end{aligned}

    The quadratic is not factorable. Set each factor equal to zero. First you have w = 0. Then, for the quadratic, use the quadratic formula:
    \begin{aligned}<br />
w &= \frac{5 \pm \sqrt{(-5)^2 - 4(1)(-2)}}{2(1)} \\<br />
w &= \frac{5 \pm \sqrt{25 + 8}}{2} \\<br />
w &= \frac{5 \pm \sqrt{33}}{2} \\<br />
 \end{aligned}

    So there are three answers
    \begin{aligned}<br />
w &= 0 \\<br />
w &= \frac{5 + \sqrt{33}}{2} \\<br />
 w &= \frac{5 - \sqrt{33}}{2}<br />
 \end{aligned}
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