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.

Question

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.

Answer 1

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.

Answer 2

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