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nicok06
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Root, Square and absolute value 30 Jul 2015, 02:03
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\(\sqrt{x^{2}} = -x\)

What are possible values for x?

By thinking logically one finds that x=0 or x x has an infinite number of solutions

2) \(x^{2} = - (x)^{2}\)
-> x = 0


What did I do wrong? Thanks :)



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SKMishra
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Root, Square and absolute value 30 Jul 2015, 04:24
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nicok06
\(\sqrt{x^{2}} = -x\)

What are possible values for x?

By thinking logically one finds that x=0 or x<0

However, I want to find an algebraic way to solve this equation, so I started like this:

\(\sqrt{x^{2}} = -x\)
\(|x^{2}| = (-x)^{2}\)
\(|x^{2}| = (x)^{2}\)

1) \(x^{2} = (x)^{2}\)
-> x has an infinite number of solutions

2) \(x^{2} = - (x)^{2}\)
-> x = 0


What did I do wrong? Thanks :)
Show more

Always avoid squaring an equation, because it may add extra solutions to the equation. For example, say x = 3, i.e. we definitely know that x has only one value which is equal to 3. However, if you square the equation, then you get \(x^{2} = 9\), which will have two solutions x = 3 and x = -3. An extra solution (x = -3) is getting added because of squaring.

One way to simplify this equation is:

\(\sqrt{x^{2}} = |x|\)
Also, as per the definition of absolute value, |x| = x (for \(x\geq{0}\)), and |x| = -x (for x < 0).

Hence, the \(\sqrt{x^{2}} = |x| = -x\) will hold true for negative values of x.

UPDATE: Also, please note that \(\sqrt{x^{2}} = |x|\) and \(-{\sqrt{x^{2}}} = -|x|\).

Reference: GMAT Official Guide 13th edition, page 114 (section 4.0 Math Review, sub-section 7. Powers and Roots of Numbers).
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Root, Square and absolute value 02 Aug 2015, 22:53
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A key step in this problem is to fully understand why \(\sqrt{x^2} = |x|\). Once you are comfortable with that, finding the solution to the above problem becomes easy.

Cheers,
Dabral



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