Question 132098
{{{n}}}= number of poles in a km
{{{x }}}= initial distance between poles in meters
(1) {{{n = (1000/x) + 1}}} add 1 to account for poles at start and finish
(2) {{{n - 5 = 1000/(x+10) + 1}}}
Add {{{5}}} to both sides of (2)
{{{n = 1000/(x+10) + 6}}}
Set the right side equal to right side of (1)
{{{1000/(x+10) + 6 = 1000/x + 1}}}
multiply both sides by {{{x*(x+10)}}}
{{{1000x + 6x*(x+10) = 1000*(x+10) + x*(x+10)}}}
{{{1000x + 6x^2 + 60x = 1000x + 10000 + x^2 + 10x}}}
{{{5x^2 + 50x - 10000 = 0}}}
divide through by {{{5}}}
{{{x^2 + 10x - 2000 = 0}}}
complete the square
{{{x^2 + 10x + (10/2)^2 = 2000 + (10/2)^2}}}
{{{(x + 5)^2 = 2025}}}
{{{x = 45 - 5}}}
{{{x = 40}}} m between poles
check answer:
(1) {{{n = (1000/x) + 1}}} 
(2) {{{n - 5 = 1000/(x+10) + 1}}}
---------------------------------
(1) {{{n = (1000/40) + 1}}}
{{{n = 26}}}
---------------------------------
(2) {{{n - 5 = 1000/(40+10) + 1}}}
{{{n - 5 = 21}}}
{{{n = 26}}}
OK