Question 276270
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Let *[tex \Large x] represent one of the integers.


Let *[tex \Large y] represent the other one.


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x\ -\ y\ =\ 6]


and


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x^2\ -\ y^2\ =\ 132]


Solve the first equation for either of the variables.  I choose to solve for *[tex \Large y]:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ x\ -\ 6]


Substitute the expression in *[tex \Large x] that is equal to *[tex \Large y] for *[tex \Large y] in the second equation:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x^2\ -\ \left(x\,-\,6\right)^2\ =\ 132]


Expand and simplify:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x^2\ -\ \left(x^2\,-\,12x\,+\,36\right)\ =\ 132]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x^2\ -\ x^2\ +\ 12x\ -\ 36\ =\ 132]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 12x\ -\ 36\ =\ 132]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 12x\ =\ 168]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ 14]


And then the other number must be 8


Check:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 14\ -\ 8\ =\ 6]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 14^2\ -\ 8^2\ =\ 196\ -\ 64\ =\ 132]



John
*[tex \LARGE e^{i\pi} + 1 = 0]
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