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# help

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When 125 is added to a number, it becomes a perfect square. When 176 is added to that same number, it also becomes a perfect square. What is the number?

Feb 23, 2020

#1
+1

a^2 - b^2  =  ([176+ n] - [125 +n])

a^2  - b^2  =  176 - 125

a^2 - b^2  =  51

Note  that   if   a  =10   and b = 7  this is true  because     10^2  - 7^2  =  100 - 49  - 51

So  a^2 =  100     and  b^2 =  49

So

100 - 176   =  n  =   -76

49  - 125  = = n  = -76

So....the  number  is   -76   Feb 23, 2020
#2
+1

Another way to do this problem:

x + 125  =  a2                                                        x + 176  =  b2

x + 176  =  b2                                                       x + 125  =  a2

Subtracting the first equation from the second:           51  =  b2 - a2

factoring:        17·3  =  (b + a)(b - a)

This means that  b = 10  and  a = 7

So:  x + 125  =  a2      --->     x + 125  =  72     --->     x + 125  =  49     --->     x  =  -76

x + 176  =  b2     --->     x + 176  =  102    --->     x + 176  =  100   --->     x  =  -76

Feb 23, 2020
#3
+2

When 125 is added to a number, it becomes a perfect square. When 176 is added to that same number, it also becomes a perfect square. What is the number?

500 +125 = 625       sqrt(625) = 25

500 + 176 = 676      sqrt(676) = 26 Feb 23, 2020