First root problems

My son had bit of a struggle with problem #18 from the 2007 AMC 10 b today. Here’s a link to the problem:

http://www.artofproblemsolving.com/wiki/index.php?title=2007_AMC_10B_Problems#Problem_18

and here’s the problem itself:

Tonight we took a closer look at this problem.  He was able to solve it, but during the solution we had an interesting detour caused by the square root of 2.

Here are his introductory thoughts about the problem as well has his first steps toward the solution:

Next we went on to solve the equation he came up with in the last video:

$\sqrt{2} x = x + 2$

His idea for how to solve this equation is to square both sides:

After finding the answer to the equation, we went back to the original linear equation and found other ways to solve it. My son’s solution starting around 1:05 and going for maybe 45 seconds was fascinating to me. I hope that after this short little additional talk that he is more comfortable solving linear equations.

So, a challenging problem for sure, but it was nice to stumble on the little algebra trouble, too. Hopefully a little extra practice here and there will help him gain a better understanding of both algebra and geometry.