Is the following reasoning right?

Question

I saw this post on Reddit and it had the following comment

√x²=|x| It's the principle square root solution. So the square root of 49 is 7. It's the definition of square root. However the equation x²=7² has two solutions +7 & -7 Only because x²=7² --> x²-7²=0 --> (x+7)(x-7)=0 -->x= ±7 That's the difference between the two. Here's an intuitive example to understand that: y²=x is not the same as y=√x On plotting the points the first equation will result in a complete parabola whereas, in the second equation it would make a half parabola because the function is defined in such a way.

Is this reasoning right? Because as far as I can remember we always wrote the value of a square root in ± form

Answer 1

To clarify this: Take the equation $$y=x^2$$ for $y>0$. Then, this equation has two solutions, $y=\sqrt x$ and $y=-\sqrt x$. This is what you mean by "$\pm$ form".

The square root is defined as the non-negative solution of this equation. Simply because we want the square root to have specific properties (like that it becomes a function again), so we have to give the square root a unique value.

You can also think of it as the inverse function of the function $f(x)=x^2$. Because $f$ has to be bijective, we somehow have to restrict the domain; to get the same as above, we can choose $f:\mathbb R_{\geq0}\to\mathbb R_{\geq0}$. This also has the implication that the square root is only the non-negative solutuion of the equation above.

To sum this up, there is a difference between the square root (which is unique) and the solutions of quadratic equations (which do not have to be unique).

Answer 2

I get why you're confused here. But the idea that $y^2 = x$ is not equivalent to saying $y =\sqrt x$ hits the nail.

$$y^2 = x$$

has two solutions, $\pm \sqrt x$ as squaring both quantities will give me $x$. The solution to this equation are called the "square roots".

Now when I look at say

$$y = \sqrt{x^2}$$.

then $y = |x|$ is the only solution, and this solution is called the "principal square root".

"But why?" you can ask. "If I have $y = \sqrt{49}$, I can write $49 = (-7)^2$ and $49=(7)^2$, right? So that means both $y=\pm 7$ satisfy the equation $y=\sqrt{49}$?" you can implore.

Good question!

Now you see the symbol $\sqrt .$ really means the principal square root, i.e. only the positive counterpart. It's in the definition of $\sqrt .$ that we only take the positive solution.