A little question to refresh your mind

Algebra Level 3

If $\sqrt{n-\sqrt a} + \sqrt{n+ \sqrt a} = \sqrt b$ and $a=mb$ , where $n,a$ and $b$ are distinct positive integers with $n\geq 3$ , find the value of $m$ .

\frac12

2^{-1/2}

2

\sqrt2

1 solution

James Wong
Mar 24, 2016

$√(n-√a) +√(n+√a) =√b$
By squaring both sides and arranging terms,
$n^2-a=(((b^2)/2)-n)^2$
$bn-(b^2)/4=a$
Substitute $a=mb$ into the equation and simplify it,
$n=m+b/4$
So, for $n$ and $b$ to be positive integers, only $2$ or $1/2$ can be the answer.
For $m=2$ , the smallest number of $b$ is $4$ such that $n=3,a=8$ .
Substitute $n=3,a=8$ into the original equation, we get $b=8$ which is not equal to $4$ .
So, $m=2$ is wrong.
For $m=1/2$ , the smallest number of $b$ is $10$ such that $n=3 a=5$ .
Substitute $n=3,a=5$ into the original equation, we get $b=10$ which is the same as we predicted,
So, $m=1/2$

A little question to refresh your mind

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