Question 831570
Their departure point and their positions at any time form a right triangle with the right angle at the departure point.  Pythagorean Theorem governs the triangle's sides.  


Rate*Time=Distance, for any single direction.


East boat:  (r+7)*2 miles
South boat: r*2 miles
Hypotenuse, distance between boats at 2 hours:  34 miles.
We do not know but want to find r, the speed of the southbound boat.


{{{(2^2)(r+7)^2+(r*2)^2=34^2}}}, just according to the triangle and Pythagorean Theorem.
{{{4(r^2+14r+49)+4r^2=34^2}}}
{{{4r^2+56r+196+4r^2=1156}}}
{{{8r^2+56r+196-1156=0}}}
{{{8r^2+56r-1060=0}}}
{{{highlight_green(2r^2+14r-265=0)}}}
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Discriminant:  14^2-4*2(-265)=2316=2*1158=2*2*579=2*2*3*193
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{{{r=(-14+sqrt(2316))/(2*2)}}}
{{{r=(-14+sqrt(2*2*3*193))/(2*2)}}}
{{{r=(-14+2*sqrt(3*193))/(2*2)}}}
{{{r=(-7+sqrt(3*193))/2}}}
{{{highlight(r=(-7+sqrt(579))/2)}}}
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{{{r=8.5}}} approximately, mph