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Math Help - Need help understanding how to factor out of a complex root

  1. #1
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    Question Need help understanding how to factor out of a complex root

    In the process of brushing up on my math skills, I came across an example where \sqrt{x^2+x}+x has x factored out, resulting in x(\sqrt{1+\frac{1}{x}}+1).

    I am confused as to how this is accomplished, and would greatly appreciate it if someone could enlighten me.


    I tried looking over various texts on dealing with exponents, but none of them cover this type of issue.
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  2. #2
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    \sqrt{x^2+x}+x

    \sqrt{x(x+1)}+x

    \sqrt{x}\sqrt{x+1}+x

    x^{\frac{1}{2}}\sqrt{x+1}+x

    x\times x^{\frac{-1}{2}}\sqrt{x+1}+x

    x\left( x^{\frac{-1}{2}}\sqrt{x+1}+1\right)

    You can simplify the expression inside the brackets to finish.
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  3. #3
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    Quote Originally Posted by Lancet View Post
    In the process of brushing up on my math skills, I came across an example where \sqrt{x^2+x}+x has x factored out, resulting in x(\sqrt{1+\frac{1}{x}}+1).

    I am confused as to how this is accomplished, and would greatly appreciate it if someone could enlighten me.


    I tried looking over various texts on dealing with exponents, but none of them cover this type of issue.
    for x > 0 ...


    \sqrt{x^2+x} + x =

    \displaystyle \sqrt{x^2\left(1+\frac{1}{x}\right)} + x =

    \displaystyle \sqrt{x^2} \cdot \sqrt{1+\frac{1}{x}} + x =

    \displaystyle x \cdot \sqrt{1+\frac{1}{x}} + x =

    \displaystyle x\left(\sqrt{1+\frac{1}{x}} + 1\right)
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  4. #4
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    Thank you both. I see where the missing techniques are.
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