Pedantic On Mathematical Software (Middle School Algebra)

17:37:00  

This time Maple get it wrong while Wolfram get it flawlessly. Consider the following middle-school level problem:

Solve the following equation on $x$

\[\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=1\]

Solution:

The equation is equivalent to $\sqrt{\left(\sqrt{x-1}-2\right)^2}+\sqrt{\left(\sqrt{x-1}+3
\right)^2}=1$ so that we have

\[\left|\sqrt{x-1}-2\right|+\left|\sqrt{x-1}-3\right|=1\]

We divide into three cases:

i) When $\sqrt{x-1}< 2$ then we have $\sqrt{x-1}< 3$ so that  the above equation becomes
\[2-\sqrt{x-1}+3-\sqrt{x-1}=1\]
that is $\sqrt{x-1}=2$, contradicting $\sqrt{x-1}< 2$.

ii) When $\sqrt{x-1}> 3$ then we have $\sqrt{x-1}> 2$ so that the equation becomes
\[\sqrt{x-1}-2+\sqrt{x-1}-3=1\]
that is $\sqrt{x-1}=3$, contradicting $\sqrt{x-1}> 3$.

ii) When $2\leq \sqrt{x-1}\leq 3$ then we have
\[\sqrt{x-1}-2+3-\sqrt{x-1}=1\]
which is equivalent to $1=1$, and so all $x$ on this case are the solutions, those are $x\in [5,10]$.

However, if you input the command

\[solve(\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}-1,x)\]

into Maple 16, it gives you an output

\[x=x\]

Other math  software like Wolfram Mathematica gives us the correct answer $5\leq x \leq 10$.

Using the method of squaring you'll get $1=1$ as well.