What is the value of y?

BrentGMATPrepNow could you please explain the above notation effect, if the equation was \(\sqrt{x^2} = 19\) ;

Typically, the solution would be as follows

If \(\sqrt{x^2} = 19\)

then \(|x|\) = 19

i.e. \(x\) = +19 or -19

but considering the impact of the notation, should we consider only the positive value of \(x\) i.e \(x\) = +19 ?

I am unsure about proving sufficiency by my understanding of the solutions here and from another similar problem linked below
https://gmatclub.com/forum/what-is-the-value-of-x-1-x-2-19-2-19-2-x-348768.html

It would really help if you can explain by considering the perspective of both the problems.

My understanding of why \(\sqrt{x^2} = 19\) can take both positive and negative values of \(x\) is because we have an even power(square) within the radical, therefore, the value under the radical is always positive. Whereas in the case of \(19 = \sqrt{x}\), \(x\) can take only positive values as square root of negative number is not a real number.

Is the above approach correct and valid for determining sufficiency?

Thanks
RP

Some students make the mistake thinking that the square root of an number gives a positive and negative answer.
No, a square root of a number occupies only one point in the real number line.
The statement 1 results in two number because the statement presents an incomplete quadratic equation that is a parabola with Δ > 0 with two real roots.

The square root notation tells to take only the positive root.
So, the equation \(\sqrt{x^2} = 19\) follows this rule because it's not suggesting that \(\sqrt{x^2} = -19\)