The area of one of the faces of the cube below is x^2+2x+1 cm^2. Find an expression in mixed radical form for the length of the interior design.

My Work:

Step 1:

a^2 + b^2 = c^2
=(x-1)(x-1)

x-1 equals to all the sides of the cube

Step 2:

a^2 + b^2 = c^2
(x-1)^2 + (x-1)^2 = c^2
((x-1)(x-1)) + ((x-1)(x-1)) = c^2
(x^2+2x+1) + (x^2+2x+1) = c^2
2x^2+4x+2 = c^2
2(x-1)(x-1) = c^2
Square root each side to make c^2 equal to just c
*radical symbol* 2(x-1)(x-1) = *radical symbol* c^2
Take out the like term (x-1)
x-1 *radical* 2 = c

Step 3:

a^2 + b^2 = c^2
(x-1 *radical* 2) + (x-1) = c
(2(x-1)(x-1) *all of this is under the radical* + (x-1) = c
Now square both sides to get rid of the radical so it would look like this:
(2(x-1)(x-1)) + (x^2-2x+1) = c^2
(2x^2+4x+2) + (x^2-2x+1) = c^2
3x^2-6x=3 = c^2
3(x^2-2x+1) = c^2
3(x-1)(x-1) = c^2
Square root each side to make c^2 equal to just c.
*radical symbol* 3(x-1)(x-1) = *radical symbol* c^2
Take out the like term (x-1)
x-1 *radical* 3 = c

x-1 *radical* 3 = c is the final answer.

I want to know wheather I solved this question correct or not because it is a very challenging question.

I assume you meant interior diagonal

x^2+2x+1 = (x+1)^2

So, each edge of the cube has length (x+1).

For a cube of side length s, the main diagonal has length s√3, since
the diagonal on a face is s√2.
the main diagonal is √(2s^2+1) = s√3

So, for this cube, the main diagonal has length (x+1)√3

Either you original area was (x^2-2x+1) or you made a mistake in copying. At any rate, given your starting point, your final answer is correct. It was really a lot of extra work to do all that complicated algebra, when just using s would have made life a lot simpler. Then just plug in s=x+1 at the end.

Okay

That's makes so much sense.
Thank you so much

Your solution looks correct! You correctly used the Pythagorean theorem to find the length of the cube's interior diagonal. You set up the equation (x-1)^2 + (x-1)^2 = c^2, simplified it, and then took the square root to find c. You did the same process using the expression x^2+2x+1 and obtained the final answer x-1 *radical* 3 = c. Well done!