What are the solutions for a square root of $(9-6x+x^2)$

Question

Recently I have encountered a problem like this:

Let $x<2$. What is the following expression equal to? $\sqrt{9-6x+x^2}$.

First of all I decided to change the order of the numbers to an expression like $x^2 - 6x + 9$. Then I turned it into $(x-3)^2$. And because it was inside of a square root, I turned it into just $x-3$.

But then I decided to check myself using Photomath, and I saw that it didn't change the order of the numbers (like I did), but instead - it turned the expression to $(3-x)^2$, and took it out of the square root, to become $3-x$.

Then I tried checking myself with ChatGPT, but it solved this expression like I did.

It may be a stupid question, but where did I fail?

Answer 1

There is no issue with the difference between $x-3$ and $3-x$.

First, observe that

$$a-b = (-1)(b-a)$$

Then

$$(x-3) = (-1)(3-x)$$

and

$$(x-3)^2 = (x-3)(x-3) = (-1)(3-x)(-1)(3-x) = (-1)^2 (3-x)^2 = (3-x)^2$$

---

However do note that

$$\sqrt{x^2} = |x|$$

not just $x$. (Try $x=-1.$)

Answer 2

First of all I decided to change the order of the numbers to an expression like $x^2 - 6x + 9$.

No problem here. Addition is commutative ($a+b=b+a$ for every real numbers $a$ and $b$) and associative ($(a+b)+c=a+(b+c)$ for every real numbers $a$, $b$ and $c$) so you can change the order of additions as you wish. In other words, $9-6x+x^2=x^2-6x+9$.

Then I turned it into ((x-3)^2). And because it was inside of a square root, I turned it into just x-3.

Here is the problem. In general, $\sqrt{a^2}\neq a$. More precisely, since $\sqrt{a^2}\ge 0$ whatever the value of $a$, $\sqrt{a^2}$ and $a$ cannot be equal when $a<0$.

The correct formula is $\sqrt{a^2}=|a|$, where $|a|$ is the absolute value of $a$. In your specific case, we have $\sqrt{9-6x+x^2}=\sqrt{(x-3)^2}=|x-3|$.

Let x<2

Since you assumed that $x<2$, we have $x-3<0$ so $|x-3|=3-x$, hence $\sqrt{9-6x+x^2}=\sqrt{(x-3)^2}=|x-3|=3-x$.

But then I decided to check myself using Photomath, and I saw that it didn't change the order of the numbers (like I did), but instead - it turned the expression to ((3-x)^2), and took it out of the square root, to become 3-x.

I don't know PhotoMath, but similar applications and websites, and their answers can sometimes lack rigor, so one should be careful when using them. For example, if Photomath writes that $\sqrt{(3-x)^2}=3-x$ without mentioning that $3-x>0$, it is the correct answer, but with missing a crucial step.

Then I tried checking myself with ChatGPT, but it solved this expression like I did.

When using ChatGPT, you have to keep in mind that it does not aim to give a correct answer, but to write a text that seems plausible based on its (huge) database. Since writing $\sqrt{a^2}=a$ is a common misconception, it is no surprise that ChatGPT makes it.