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The difference between the square roots of a number is 30. What is the number?

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4 Answers

Grigori:
 
The difference between the square roots of a number is 30. What is the number?

Analytical approach.

Let the required number be denoted by x2.  The square root of x2 is (+ or -) x. If you take the second number as + x (that is x itself) the difference is  x - x = 0, not 30. So that you must choose the second number as - x. The difference between + x and - x is equal to x - (-x) = 2x. We are given that this difference is 30. Therefore, 2x = 30. That is x = 15, and the required number is 152 = 225.

Back check: Square roots of 225 are +15 and -15. Their difference is 15 – (15) = 30. Hurray!
 
Dattaprabhakar (Dr. G.)
 
P.S. Generalization (for adavnced students): If M is any real number, the number (M/2)2 is such that the difference between its square roots is M.  Proof: On request.  Post a comment if you rrrrrreeeeeallllllyyyyy want to see it.
Carlos, remember that any number has two square roots, a positive one and a negative one.  For example, the square roots of 25 are -5 and 5, the difference of which is 10.  Difference is how far apart two numbers are.  
 
The question is asking which number has square roots that have a difference of 30.  The two square roots would be the "same" but one negative and one positive, 30 apart.
Interestingly, you can do this one logically just as well and as easily as Grigori's mathematical method (it is, in fact, the same thing, just in a different idiom):
 
The square root is defined as the number which, when multiplied by itself, gives the number in question (x in this problem).  In most cases, the number in question will be a positive one.  All positive numbers have two square roots, one positive and one negative.  This is because multiplying a positive by a positive results in a positive and multiplying a negative by a negative results in a negative.  Because the positivity/negativity of the roots, in a way doesn't matter, we can see that the absolute values of the two (that is, their value without regard to negativity/positivity) must be the same.  Thus, they must reflect around zero, each being an equal distance away from zero.  Thus, since we know the difference between the roots is 30, half of which is 15, we must therefore also know that the roots are going to be 15 and -15, so we can just go ahead and square that.
This means one square root is positive while another one is negative. 
Let x be the number you are looking for. Your word statement can be written as the following equation
 
                      √x - (-√x) = 2√x = 30  or 
 
                    √x = 15    and x = 152 = 225.
 
 Answer: x = 225