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Squaring both sides of an equality

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T4Star
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Squaring both sides of an equality [#permalink] New post   20 Mar 2024, 10:57
Hi,

Something simple has been confusing me. I understand you dont square two sides on an inequality unless you know both sides are non-negative. But is the same condition also applicable for equality? Or can we square straight away on both sides?

Eg: If sqrt(6-x) = x, find all possible roots of x­

Squaring both side would give me:

6-x = x^2
x^2 + x - 6 = 0
(x + 3) (x - 2) = 0

So, x= -3 or x = 2

But if I plugin x = -3 to original equation I get sqrt (9), which is equal to 3 and not -3 (since square root of a constant is always positive)
So, x = -3 is not a valid solution?

bb, Bunuel
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Squaring both sides of an equality [#permalink] New post   20 Mar 2024, 11:08
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T4Star wrote:
Hi,

Something simple has been confusing me. I understand you dont square two sides on an inequality unless you know both sides are non-negative. But is the same condition also applicable for equality? Or can we square straight away on both sides?

Eg: If sqrt(6-x) = x, find all possible roots of x­

Squaring both side would give me:

6-x = x^2
x^2 + x - 6 = 0
(x + 3) (x - 2) = 0

So, x= -3 or x = 2

But if I plugin x = -3 to original equation I get sqrt (9), which is equal to 3 and not -3 (since square root of a constant is always positive)
So, x = -3 is not a valid solution?

bb, Bunuel

x being equal to the square root of a number means that x cannot be negative because the square root, or any even root for that matter, cannot produce a negative result. Therefore, if you square, you should plug back to verify the validity of the roots.­ So, \(\sqrt{6-x} = x\) has only one root: x = 2.

 ­
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Re: Squaring both sides of an equality [#permalink] New post   20 Mar 2024, 11:53
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Hi. I am a bit rusty these days but I’m wondering if you have run into this situation on the GMAT questions.

It’s great you have an example but it doesn’t sound like a GMAT type of an example so I’m trying to see if this is something from math or from GMAT quant. The small differences between the two and obviously big differences in scope.

Because as you know, on the GMAT we only use the positive root (as you mentioned) so I’m wondering if this is a type of a question that GMAT test writers would run away from…

PS. But I am genuinely curious if perhaps this is something you encountered on the GMAT or in the GMAT textbook.

Posted from my mobile device
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Re: Squaring both sides of an equality [#permalink] New post   21 Mar 2024, 07:23
­Thanks, Bunuel. So, I guess you really have to check if the solution is in agreement with the original questio.

bb, I haven't come across this example on a GMAT resource. I was thinking about situations in which squaring both sides of the equation can be troublesome - and came up with this example.
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Re: Squaring both sides of an equality [#permalink]
21 Mar 2024, 07:23
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