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Question 488335: I am having an issue with the following problem:
-√5-2u = 21
5-2u is all underneath the square root symbol.
A hint tells me to square both sides, but I can't get that to work.
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Let's try to use the hint so you can find out where you might be having difficulty. You are given:
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Before we begin, take note of the fact that this equation has a negative sign on the left side, but the right side is positive. For this equation to be true, the answer to taking the square root of (5 - 2U) must be negative so the negative sign in front of the radical makes the entire left side positive like the right side is. More on this later.
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Let's square both sides:
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Note that on the left side I've included the minus sign that precedes the square root radical. It (the minus sign) gets squared also, so after squaring, it becomes a plus sign.
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I strongly suspect that this may be the source of your problem.
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Also recall that when you square a quantity that is under the square root radical, you just get the quantity itself. For example:
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 and 
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So for the problem you were given when the entire left side is squared (both the negative sign and the radical) it becomes:
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and the when the right side  is squared it becomes  . So after both sides are squared the equation becomes:
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Subtract 5 from both sides:
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Which simplifies to:
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To solve for U divide both sides by -2 to get:
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And after dividing out the left side we get the answer:
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Let's check the answer by substituting it into the original equation:
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Start with:
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Substitute -218 for U to get:
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Do the multiplication of -(2)*(-218) = + 436 under the radical:
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Add the two terms under the radical on the left side and you get:
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When you take the square root of 441 you can consider two possible answers: plus 21 and minus 21. (Normal convention is that the square root radical will give just positive answers.) However, if you square either +21 or -21 the answer will be +441. Since we have the minus sign before the radical, in order to get the equation to be true, we must use the -21 answer and discard the +21 answer. With that understanding, the equation will balance using U = -218.
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The critical point to be made is that with U = -218 we can still get the equation to be true, but to do so we have to consider ignore the positive answer created by the square root radical and allow for negative answers to be considered. Barring that, this problem cannot be solved for U. The original source of the problem with the negative sign occurs because in solving for U we needed to square the minus sign when we squared the entire left hand side.
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The problem really gets "messy" if you don't square the minus sign (-1) when you square the entire left side of the original problem. If you square just the radical and not the minus sign in front of the radical, the left side squared becomes:
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 which becomes 
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When you set that equal to the right side squared the equation becomes:
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Add +5 to both sides and this simplifies to:
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and dividing both sides by 2 the answer becomes:
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In checking this answer by substituting +223 for U in the original equation the term under the radical becomes  which simplifies to a negative number 5 - 446 = -441. Until you learn about complex numbers (real and imaginary parts), taking the square root of a negative number is not possible. Certainly it does not apply to this problem since the right side of the equation (21) is a real number. This difficulty with handling the negative sign may have been the source of your difficulty.
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Hope this helps you ...
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