Question 199563
Remember, the distance-rate-time formula is 


{{{d=rt}}}


Since he "traveled 150 miles at a certain speed", this means that the first equation is {{{150=rt}}} (simply plug in d=150)


The statement "Had gone 20mph faster, the trip would have taken 2 hour less", tells us that the next equation is {{{150=(r+20)(t-2)}}} (the distance is the same, but the speed is now 20 mph faster and the time is 2 hours shorter)



The goal is to use this system to solve for "r" (and if we want, "t" also)



{{{150=rt}}} Start with the first equation.



{{{150/r=t}}} Divide both sides by r.



{{{t=150/r}}} Rearrange the equation



{{{150=(r+20)(t-2)}}} Move onto the second equation.



{{{150=(r+20)(150/r-2)}}} Plug in {{{t=150/r}}} 



{{{150=150-2r+3000/r-40}}} FOIL



{{{150r=150r-2r^2+3000-40r}}} Multiply EVERY term by the LCD {{{r}}} to clear out the fraction.



{{{0=150r-2r^2+3000-40r-150r}}} Subtract {{{600r}}} from both sides.



{{{0=-2r^2-40r+3000}}} Combine and rearrange the terms.



Notice we have a quadratic in the form of {{{Ar^2+Br+C}}} where {{{A=-2}}}, {{{B=-40}}}, and {{{C=3000}}}



Let's use the quadratic formula to solve for "r":



{{{r = (-B +- sqrt( B^2-4AC ))/(2A)}}} Start with the quadratic formula



{{{r = (-(-40) +- sqrt( (-40)^2-4(-2)(3000) ))/(2(-2))}}} Plug in  {{{A=-2}}}, {{{B=-40}}}, and {{{C=3000}}}



{{{r = (40 +- sqrt( (-40)^2-4(-2)(3000) ))/(2(-2))}}} Negate {{{-40}}} to get {{{40}}}. 



{{{r = (40 +- sqrt( 1600-4(-2)(3000) ))/(2(-2))}}} Square {{{-40}}} to get {{{1600}}}. 



{{{r = (40 +- sqrt( 1600--24000 ))/(2(-2))}}} Multiply {{{4(-2)(3000)}}} to get {{{-24000}}}



{{{r = (40 +- sqrt( 1600+24000 ))/(2(-2))}}} Rewrite {{{sqrt(1600--24000)}}} as {{{sqrt(1600+24000)}}}



{{{r = (40 +- sqrt( 25600 ))/(2(-2))}}} Add {{{1600}}} to {{{24000}}} to get {{{25600}}}



{{{r = (40 +- sqrt( 25600 ))/(-4)}}} Multiply {{{2}}} and {{{-2}}} to get {{{-4}}}. 



{{{r = (40 +- 160)/(-4)}}} Take the square root of {{{25600}}} to get {{{160}}}. 



{{{r = (40 + 160)/(-4)}}} or {{{r = (40 - 160)/(-4)}}} Break up the expression. 



{{{r = (200)/(-4)}}} or {{{r =  (-120)/(-4)}}} Combine like terms. 



{{{r = -50}}} or {{{r = 30}}} Simplify. 



So the <i>possible</i> solutions are {{{r = -50}}} or {{{r = 30}}} 



Since a negative speed doesn't make any sense, this means that we must ignore {{{r = -50}}}



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Answer:


So the only solution is {{{r = 30}}} which means that he was traveling 30 mph. So you are correct, good job.