Question 1027670
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If the surface of a cube is 6x^2 -36x +54, what is the expression for the volume of the cube?
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        I will give different solution from that by @mananth and from that by @Theo,

        and at the end I will explain why the solutions by both these tutors are incorrect and why they should be changed.



<pre>
The surface area of the cube is given by the polynomial expression  6x^2 - 36x + 54.


It is the combined area of 6 identical faces of the cube.  
The area of each face separately is

    (6x^2 - 36x + 54)/6 = x^2 - 12x + 9 = (x-3)^2.


Notice that the given polynomial is always non-negative, 
so its values make sense as the face area for all values of x =/= 3.


The length of the side of this cube is  {{{sqrt((x-3)^2)}}} = |x-3|.


Notice that I use the absolute value  |x-3|  here, 
as the length is always assumed to be non-negative.


The volume of the cube is then  {{{abs(x-3)^3}}}.


This formula works for all values  x =/= 3, including  x < 3, 
while the formulas in the posts by @mananth and by @Theo do not work at x < 3 (!)
giving negative values for the edge length and for the volume.


So, my solution is MORE UNIVERSAL and more ACCURATE than the @mananth' solution and/or than the @Theo' solution.
</pre>

Actually, this seemingly simple task has a &nbsp;{{{highlight(highlight(well))}}} &nbsp;{{{highlight(highlight(hidden))}}} &nbsp;murderous &nbsp;{{{highlight(highlight(trap))}}}.


My solution discloses the trap, teaches you on how to avoid this trap 
and also teaches you to be always aware and always accurate in Math.



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<H3>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;It is &nbsp;&nbsp;I N T E R E S T I N G</H3>

Today, &nbsp;on &nbsp;Feb.14,2026, &nbsp;I submitted this problem to &nbsp;GOOGLE &nbsp;AI &nbsp;Overview to see how it treats it.


Google AI Overview practically repeated incorrect/incomplete solution by @mananth and by @Theo.


Naturally, &nbsp;I reported to &nbsp;Google &nbsp;AI &nbsp;about their fault through their feedback system.



Then I made my next experiment. I submitted the same problem to other AI,
math-gpt.org , which (i think) is slightly more advanced.


This other AI repeated the same tediousness and produced the same "lame" solution 
(which reminds me a lame horse with three legs).


It is the real world of AI in the area of solving school Math problems, in which we all live now.