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Thread: How to rearrange/manipulate algebraically?

  1. #1
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    How to rearrange/manipulate algebraically?

    This is a formula for chemistry, but I'm having trouble figuring out out it was rearranged. I'd love any help/steps/explanations on how this was rearranged!

    This is the first line:
    $\displaystyle =\frac{h^2(n+1)^2\pi^2}{2mL^2}-\frac{h^2n^2\pi^2}{2mL^2}$

    It was then manipulated to this:
    $\displaystyle =(\frac{h^2\pi^2}{2mL^2})((n+1)^2-n^2)$

    I'd appreciate any help!
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  2. #2
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    Re: How to rearrange/manipulate algebraically?

    Quote Originally Posted by maybealways View Post
    This is a formula for chemistry, but I'm having trouble figuring out out it was rearranged. I'd love any help/steps/explanations on how this was rearranged!

    This is the first line:
    $\displaystyle =\frac{h^2(n+1)^2\pi^2}{2mL^2}-\frac{h^2n^2\pi^2}{2mL^2}$

    It was then manipulated to this:
    $\displaystyle =(\frac{h^2\pi^2}{2mL^2})((n+1)^2-n^2)$

    I'd appreciate any help!
    They took out the highest common factor.
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    Re: How to rearrange/manipulate algebraically?

    Factor out $\displaystyle \frac{h^2 \pi^2}{2mL^2}$.
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  4. #4
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    Re: How to rearrange/manipulate algebraically?

    Quote Originally Posted by maybealways View Post
    This is a formula for chemistry, but I'm having trouble figuring out out it was rearranged. I'd love any help/steps/explanations on how this was rearranged!

    This is the first line:
    $\displaystyle =\frac{h^2(n+1)^2\pi^2}{2mL^2}-\frac{h^2n^2\pi^2}{2mL^2}$
    Working from left to right, we see that there is a "$\displaystyle h^2$ in both terms. There is an $\displaystyle (n+1)^2$ in the first term but not the second. Likewise there is an $\displaystyle n^2$ in the second term but not the first. There is a $\displaystyle \pi^2$ in both terms. And, of course, there is $\displaystyle 2mL^2$ in the denominator of both terms. That means we can factor out $\displaystyle \frac{h^2\pi^2}{2mL^2}$ leaving $\displaystyle (n+1)^2- n^2$. Hence:
    $\displaystyle \frac{h^2\pi^2}{2mL^2}((n+1)^2- n^2)$.

    In fact, we could go a step further. Since $\displaystyle (n+1)^2= n^2+ 2n+ 1$, $\displaystyle (n+1)^2- n^2= 2n+ 1$ so we could write it as $\displaystyle \frac{h^2\pi^2}{2mL^2}(2n+1)$.

    It was then manipulated to this:
    $\displaystyle =(\frac{h^2\pi^2}{2mL^2})((n+1)^2-n^2)$

    I'd appreciate any help!
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