What Integral Values Satisfy This Radical Equation?

[anemone]

Here is this week's POTW:

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Find all integral values of $a$ such that $\sqrt{a+8-6\sqrt{a-1}}+\sqrt{a+3-4\sqrt{a-1}}=1$.

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[anemone]

Congratulations to the following members for their correct solution::)

1. MarkFL
2. kaliprasad
3. lfdahl
4. greg1313

Solution from MarkFL:

Spoiler

Let:

\(\displaystyle a-1=m^2\implies a=m^2+1\)

And we have:

\(\displaystyle \sqrt{m^2-6m+9}+\sqrt{m^2-4m+4}=1\)

Or:

\(\displaystyle |m-3|+|m-2|=1\)

We have the following 3 cases to consider:

i) \(\displaystyle 3\le m\)

\(\displaystyle m-3+m-2=1\)

\(\displaystyle m=3\)

ii) \(\displaystyle 2<m<3\)

\(\displaystyle 3-m+m-2=1\)

\(\displaystyle 1=1\)

Because this is an identity, we know all $m$ in this interval is a solution.

iii) \(\displaystyle m\le2\)

\(\displaystyle 3-m+2-m=1\)

\(\displaystyle m=2\)

Hence we find:

\(\displaystyle 2\le m\le 3\)

Square through, and since all are positive, the inequality does not change direction:

\(\displaystyle 4\le m^2\le9\)

Add through by 1:

\(\displaystyle 5\le m^2+1\le10\)

Hence:

\(\displaystyle 4\le a\le10\)

Since $a$ is integral, we conclude:

\(\displaystyle a\in\{5,6,7,8,9,10\}\)