# Equations

Show that there are exactly four distinct sets of integers which satisfy the attached equations:

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#### Solution Preview

x1+x2+x3 = 54 ......(1)

x1^2 + x2^2 + x3^2 = 1406 ......(2)

eliminate x3 from (2) using (1), we get,

>x1^2 + x2^2 + (54- x1 - x2)^2 = 1406

=> 2x1^2 + 2x2^2 + 2x1x2 - 108x1 - 108x2 + 54^2 = 1406

=> x1^2 + x2^2 + x1x2 - 54x1 - 54x2 + 27*27*2 = 703

Let x2 = k

=> x1^2 + (k - 54)x1 + (k^2 - 54k + 1458 - 703) = 0 ...(3)

=> x1^2 + (k - 54)x1 + (k^2 - 54k + 755) = 0

Compare with general quadratic eqn,

ax^2 + bx + c =

Soln.: x = [-b + or - ...

#### Solution Summary

This shows how to prove there are four distinct sets of integers that satisfy given conditions.

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