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# Root, Square and absolute value

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Root, Square and absolute value [#permalink]  30 Jul 2015, 01:03
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$$\sqrt{x^{2}} = -x$$

What are possible values for x?

By thinking logically one finds that x=0 or x<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
Jamboree GMAT Instructor
Status: GMAT Expert
Affiliations: Jamboree Education Pvt Ltd
Joined: 15 Jul 2015
Posts: 8
Location: India
GMAT 1: 680 Q51 V28
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Root, Square and absolute value [#permalink]  30 Jul 2015, 03:24
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Expert's post
nicok06 wrote:
$$\sqrt{x^{2}} = -x$$

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

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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Re: Root, Square and absolute value [#permalink]  02 Aug 2015, 21:53
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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
Re: Root, Square and absolute value   [#permalink] 02 Aug 2015, 21:53
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