MHB What Integral Values Satisfy This Radical Equation?

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    2015
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The discussion focuses on solving the radical equation involving integral values of \( a \). Participants are tasked with finding all integral solutions to the equation \( \sqrt{a+8-6\sqrt{a-1}}+\sqrt{a+3-4\sqrt{a-1}}=1 \). Several members successfully provided correct solutions, including MarkFL, kaliprasad, lfdahl, and greg1313. The thread emphasizes the importance of following the Problem of the Week guidelines for participation. The community engages in mathematical problem-solving, showcasing collaborative efforts in finding solutions.
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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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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
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Congratulations to the following members for their correct solution::)

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

Solution from MarkFL:
Let:

$$a-1=m^2\implies a=m^2+1$$

And we have:

$$\sqrt{m^2-6m+9}+\sqrt{m^2-4m+4}=1$$

Or:

$$|m-3|+|m-2|=1$$

We have the following 3 cases to consider:

i) $$3\le m$$

$$m-3+m-2=1$$

$$m=3$$

ii) $$2<m<3$$

$$3-m+m-2=1$$

$$1=1$$

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

iii) $$m\le2$$

$$3-m+2-m=1$$

$$m=2$$

Hence we find:

$$2\le m\le 3$$

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

$$4\le m^2\le9$$

Add through by 1:

$$5\le m^2+1\le10$$

Hence:

$$4\le a\le10$$

Since $a$ is integral, we conclude:

$$a\in\{5,6,7,8,9,10\}$$
 
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