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Math Help - Square root solving

  1. #1
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    Square root solving

    If x = 5 + 2√6,
    Find (x-1)/√x ............... Answer is 2√3............. Please explain the steps involved.
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  2. #2
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    Re: Square root solving

    The answer is 2√2. Prove that the square of the expression is 8. To simplify a fraction with the denominator a+b\sqrt{6}, multiply the numerator and the denominator by a-b\sqrt{6}.
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  3. #3
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    Re: Square root solving

    First, no, that is not the answer. Are you sure you copied it correctly?

    It would be possible, since you are given x, to find both x- 1 and \sqrt{x} and do the division directly. However, \sqrt{x} is a little tedious to find so I would look at \frac{(x- 12)^2}{x}. x= 5+ 2\sqrt{6} so x- 1= 4+ 2\sqrt{6} and (x- 1)^2= 16+ 16\sqrt{6}+ 24= 40+ 16\sqrt{6}. Then \frac{(x- 1)^2}{x}= \frac{40+ 16\sqrt{6}}{5+ 2\sqrt{6}}. Do that division by multiplying both numerator and denominator by 5- 2\sqrt{6}
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  4. #4
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    Re: Square root solving

    Hello, Amap!

    There is a typo . . .


    \text{If }x \,=\, 5 + 2\sqrt{6},\,\text{find }\frac{x-1}{\sqrt{x}}

    \text{Answer: }\,2{\color{red}\sqrt{2}}

    Note that: . 5 + 2\sqrt{6} \:=\:(\sqrt{3}+\sqrt{2})^2


    Then: . \frac{x-1}{\sqrt{x}} \;=\;\frac{(5+2\sqrt{6}) -1}{\sqrt{3}+\sqrt{2}} \;=\;\frac{4+2\sqrt{6}}{\sqrt{3} + \sqrt{2}}


    Rationalize: . \frac{4+2\sqrt{6}}{\sqrt{3}+\sqrt{2}}\cdot {\color{blue}\frac{ \sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}}} \;=\; \frac{4\sqrt{3} - 4\sqrt{2} + 2\sqrt{18} - 2\sqrt{12}}{3 - 2}

    . . . . . . . . . =\;4\sqrt{3} - 4\sqrt{2} + 6\sqrt{2} - 4\sqrt{3} \;\;=\;\;\boxed{2\sqrt{2}}
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  5. #5
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    Re: Square root solving

    Square root solving-alternate-proof.png
    Quote Originally Posted by Amap View Post
    If x = 5 + 2√6,
    Find (x-1)/√x ............... Answer is 2√3............. Please explain the steps involved.
    The answer is 2 square-root 2
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