thealpine wrote:

I heard this on a teaching platform, that whenever the GMAT gives your the square root on the prompt, you can/it is okay to only consider positive values. However, if the square root is something you come across while solving for a problem then you have to consider both positive and negative values.

Heard this from MikeMcGarry

Thoughts on this?

Indeed, it is correct to consider only positive values for the square roots on the prompt on GMAT; otherwise, you would be handling with complex roots.

It is important to bear in mind that while a "regular square root" and a "square root of some variable" may seem to be the same case, there is a difference.

Numbers have a real and an imaginary part - which is often ignored by us - but in this specific case the imaginary part plays an important role.

While the "regular square root" has an \(i^0\) form, the variable solving find positive numbers from both \(i^0\) and a combination of \(i^2\) forms - which will multiply as a \(i^4\) -, as the imaginary part follows the following power pattern:

\(i^0 = 1\)

\(i^1 = i\)

\(i^2 = – 1\)

\(i^3 = i^2 * i = (–1) * i = –i\)

\(i^4 = i^2 * i^2 = (–1) * (– 1) = 1\)

...

Having that in mind, as you may have realized, we can be sure that \(√1 = 1\), since it is in the \(√(1*i^0)\) form. On the hand, there is no certain when it comes to \(x^2=1\) as it could be both:

\((1*i^0)^2 = (1*1)^2 = 1^2 = 1\)

or

\((1*i^2)^2 = (1*-1)^2 = (-1)^2 = 1\)

Hope it helps!

As for the problem, I would just consider that 1 to any power equals 1. Hence, the given expression would be the same of \(√1\) and we could simply do

\((3–2x)+(3–2x)^2 = 1 + 1^2 = 2\)

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