Selection Examination

Question

Determine all values of the integer v for which the number v^2 + 10v + 160 is a perfect square.

Accepted Answer

Let v^2 + 10v + 160 = ^2, where is a positive integer. Then we have: align* v^2 + 10v + 160 = ^2 & (v+5)^2 + 135 = ^2 ^2 - (v+5)^2 = 135 & ( - |v+5|)( + |v+5|) = 135 = 1 3^3 5, where > |v+5| & cases - |v+5| = 1 + |v+5| = 135 cases or cases - |v+5| = 3 + |v+5| = 45 cases &or cases - |v+5| = 5 + |v+5| = 27 cases or cases - |v+5| = 9 + |v+5| = 15 cases & cases = 68 |v+5| = 67 cases or cases = 24 |v+5| = 21 cases or cases = 16 |v+5| = 11 cases or cases = 12 |v+5| = 3 cases align* arrayl arrayl = 68 v + 5 = 67 array or arrayl = 24 v + 5 = 21 array or arrayl = 16 v + 5 = 11 array or arrayl = 12 v + 5 = 3 array [1.5ex] arrayl = 68 v = 62 or v = -72 array or arrayl = 24 v = 16 or v = -26 array [1.5ex] or arrayl = 16 v = 6 or v = -16 array or arrayl = 12 v = -2 or v = -8 array . array Therefore v…