Question 173348
You're off to a great start. I'll start where you left off.




{{{90r = 8r^2 - 72}}} Start with the given equation



{{{0 = 8r^2 - 72-90r}}} Subtract 90r from both sides.



{{{ 0= 8r^2-90r - 72}}} Rearrange the terms.



Notice we have a quadratic equation in the form of {{{ar^2+br+c}}} where {{{a=8}}}, {{{b=-90}}}, and {{{c=-72}}}



Let's use the quadratic formula to solve for r



{{{r = (-b +- sqrt( b^2-4ac ))/(2a)}}} Start with the quadratic formula



{{{r = (-(-90) +- sqrt( (-90)^2-4(8)(-72) ))/(2(8))}}} Plug in  {{{a=8}}}, {{{b=-90}}}, and {{{c=-72}}}



{{{r = (90 +- sqrt( (-90)^2-4(8)(-72) ))/(2(8))}}} Negate {{{-90}}} to get {{{90}}}. 



{{{r = (90 +- sqrt( 8100-4(8)(-72) ))/(2(8))}}} Square {{{-90}}} to get {{{8100}}}. 



{{{r = (90 +- sqrt( 8100--2304 ))/(2(8))}}} Multiply {{{4(8)(-72)}}} to get {{{-2304}}}



{{{r = (90 +- sqrt( 8100+2304 ))/(2(8))}}} Rewrite {{{sqrt(8100--2304)}}} as {{{sqrt(8100+2304)}}}



{{{r = (90 +- sqrt( 10404 ))/(2(8))}}} Add {{{8100}}} to {{{2304}}} to get {{{10404}}}



{{{r = (90 +- sqrt( 10404 ))/(16)}}} Multiply {{{2}}} and {{{8}}} to get {{{16}}}. 



{{{r = (90 +- 102)/(16)}}} Take the square root of {{{10404}}} to get {{{102}}}. 



{{{r = (90 + 102)/(16)}}} or {{{r = (90 - 102)/(16)}}} Break up the expression. 



{{{r = (192)/(16)}}} or {{{r =  (-12)/(16)}}} Combine like terms. 



{{{r = 12}}} or {{{r = -3/4}}} Simplify. 



So the possible answers are {{{r = 12}}} or {{{r = -3/4}}} 

  

However, a negative speed doesn't make sense. So we'll discard the possible solution {{{r = -3/4}}}




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Answer:



So the solution is {{{r = 12}}} which means that the speed of the boat in still water is 12 mph.