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# Is (sqrt) x-5^2 = 5-x?

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Director
Joined: 12 Jun 2006
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Is (sqrt) x-5^2 = 5-x? [#permalink]  10 Jun 2007, 15:07
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Is (sqrt) x-5^2 = 5-x?

1) -x|x| > 0
2) 5-x > 0

Please solve and explain stmnt. 1 in detail.

Does ((sqrt) x-5^2) * ((sqrt) x-5^2) = x-5^2?
Manager
Joined: 07 May 2007
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Are you taking the root of only x or the entire expression x-25?
Director
Joined: 12 Jun 2006
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Hi,

It's the square root of the entire left hand side expression. In other words, the entire left side of the equation is under the radical sign.
Manager
Joined: 07 May 2007
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1) -x|x| > 0 means x must be negative because |x| is always positive
So x-25 is negative and sqrt(x-25) is imaginary. On the otherhand
5-x is positive.

We can conclude sqrt(x-25) is not equal to 5-x.

1 alone is suff

2) 5-x>0 means x<5
So x-25 is negative and sqrt(x-25) is imaginary. On the otherhand
5-x is positive.
We can conclude sqrt(x-25) is not equal to 5-x.

2 alone is suff
VP
Joined: 10 Jun 2007
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You know, this is a strange one...because GMAT NEVER deals with imaginary number. If we try to solve your equation:
sqrt(x-5^2) = 5-x
=> sqrt(x-25) = 5-x
square both sides
=> x-25 = (5-x)*(5-x)
=> (x+5)*(x-5) = x^2-10x+25
=> (x+5)*(x-5) = (x-5)*(x-5)
=> (x+5) = (x-5)
=> x = x+10
We know that there is no valid solution to this problem other than imaginary number; thus, I'll go with D.
Director
Joined: 12 Jun 2006
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The OG guide offers this explanation for stmnt 1:
|x| > 0 and, since the product of -x and |x| is positive, -x>0.

Does this make sense to any of you? What am I missing? How is -x * |x| positive. Isn't |x| always +ve? That said, wouldn't:
-x * |x| = -x(x)
Manager
Joined: 23 Dec 2006
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Are you sure the left hand side is correct, and not sqrt[(x-5)^2]? It's more symmetrical than x-5^2 = x-25, so i wonder.
VP
Joined: 10 Jun 2007
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ggarr wrote:
The OG guide offers this explanation for stmnt 1:
|x| > 0 and, since the product of -x and |x| is positive, -x>0.

Does this make sense to any of you? What am I missing? How is -x * |x| positive. Isn't |x| always +ve? That said, wouldn't:
-x * |x| = -x(x)

-x|x|>0 only when x<0. For example, if x=-2, then -x|x| = 2|2| = 4. This is the same as -x>0. If you multiply -1 to -x>0, you would get x<0.
Director
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sludge wrote:
Are you sure the left hand side is correct, and not sqrt[(x-5)^2]? It's more symmetrical than x-5^2 = x-25, so i wonder.

hi sludge, yes, the entire expression, (x-5)^2 is underneath the radical sign.
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