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Math Help - Solving equation involving surds

  1. #1
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    Solving equation involving surds

    Hi,
    Any advice with the problem below would be greatly appreciated.

    X = √a + 1/√a + 1; X>0; express X - 2X in terms of a.

    btw, answer = a + 1 + 1/a; how do you derive it though?

    Thank you,
    Sean.
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  2. #2
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    Re: Solving equation involving surds

    X-2X is just -X, please recheck the question.
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  3. #3
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    Re: Solving equation involving surds

    Where you write "1/√a+ 1" do you mean (1/√a)+ 1 or 1/(√a+ 1)?
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  4. #4
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    Re: Solving equation involving surds

    Hello, Seaniboy!

    x \:=\:\sqrt{a} + \tfrac{1}{\sqrt{a}} + 1,\;x > 0.

    \text{Express }x{\color{red}^2}-2x\text{ in terms of }a.

    \text{Answer: }\:a + 1 + \tfrac{1}{a}

    x^2-2x \;=\;x(x-2)

    . . . . . . =\;\left(\sqrt{a} + \tfrac{1}{\sqrt{a}} + 1\right)\left(\sqrt{a} + \tfrac{1}{\sqrt{a}} + 1 - 2\right)

    . . . . . . =\;\bigg(\left[\sqrt{a} + \tfrac{1}{\sqrt{a}}\right] + 1\bigg)\bigg(\left[\sqrt{a} + \tfrac{1}{\sqrt{a}}\right] - 1\bigg)

    . . . . . . =\;\left(\sqrt{a} + \tfrac{1}{\sqrt{a}}\right)^2 - 1^2

    . . . . . . =\;a + 2 + \tfrac{1}{a} - 1

    . . . . . . =\;a + 1 + \tfrac{1}{a}
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  5. #5
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    Re: Solving equation involving surds

    Thanks for your help people. Apologies for the symbols in the question. I must use an 𝑥 instead of X next time!
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  6. #6
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    Re: Solving equation involving surds

    Hi, I see what you mean. I am trying to find symbols to insert to make things a little clearer. In english it is X = √a + a fraction in the form of 1 over √a + 1. Thanks for having a look at the problem anyway.
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