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Math Help - Isolating a variable

  1. #1
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    Isolating a variable

    I'm having a bit of trouble isolating "v" in the following equation:

    d = (v/g) * (v + sqrt(v^2 + 2gh ) )

    Thank you in advance for your attention and time.
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  2. #2
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    Multiply g/v to both the sides.

    \frac{dg}{v} = v + \sqrt{(v^2 + 2gh)}

    \frac{dg}{v} - v = \sqrt{(v^2 + 2gh)}

    Square both sides and simplify.
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  3. #3
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    This is precisely the part I need help with.
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  4. #4
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    I am stuck at:

    ((dv)/g - v)^2 - v^2 = 2gh

    Can I square root both sides and end up with:

    (dv)/g - v - v = sqrt(2gh) ?
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  5. #5
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    I really need help.
    What do I do after ((dv)/g - v)^2 - v^2 = 2gh ?
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  6. #6
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    (\frac{dg}{v} - v)^2 - v^2 = 2gh

    (\frac{dg}{v})^2 + v^2 -  2dg - v^2 = 2gh

    (\frac{dg}{v})^2  -  2dg  = 2gh

    Now solve for v.
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  7. #7
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    Oh, thank you. Quick question though:
    Isn't (dg)/v * -v = -dg rather than -2dg?
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  8. #8
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    No. When expanding binomial squares you will get:

    (a+b)^2= a^2 + 2ab + b^2

    Remember that you get \frac{dg}{v} \times -v = -dg as well as -v \times \frac{dg}{v} = -dg which when you add them together you get -2dg
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  9. #9
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    Wow, it has been a while. Thank you.

    Now that I have (dg)/v = sqrt(2gh + 2dg), is it possible to slimpify but turning both into their reciprocals?

    So, v/(dg) = 1/sqrt(2gh + 2dg), v = (dg)/sqrt(2gh + 2dg).

    Is this correct?
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  10. #10
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    You can further simplify it as

    v = d\sqrt{\frac{g}{2(h + d)}}
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