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

It is 8th grade homework that student nor parents can solve. It is a square root problem.  The work has to be shown.

### 4 Answers by Expert Tutors

Dattaprabhakar G. | Expert Tutor for Stat and Math at all levelsExpert Tutor for Stat and Math at all le...
5.0 5.0 (2 lesson ratings) (2)
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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.
4.9 4.9 (63 lesson ratings) (63)
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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.
Grigori S. | Certified Physics and Math Teacher G.S.Certified Physics and Math Teacher G.S.
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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.