Question 143415
Karen travels 50 miles in t hours at r miles per hour.  Carla travels 50 miles in {{{t-48/60=t-4/5}}} hours at {{{r+2}}} miles per hour.


Since {{{d=rt}}}, Karen's trip is described by {{{50=rt}}} and Carla's trip is described by {{{50=(r+2)(t-4/5)}}}


From the description of Karen's trip we can say:  {{{t=50/r}}}, so Carla's trip can be described as:


{{{50=(r+2)((50/r)-4/5)}}}
{{{50=(r+2)((250-4r)/5r)}}}
{{{250r=(r+2)(250-4r)}}}

Apply FOIL:
{{{250r=250r-4r^2+500-8r}}}


Collect terms, put in standard form:
{{{4r^2+8r-500=0}}}


Solve the quadratic:
{{{r^2+2r-125}}}


Either complete the square or use the quadratic formula.  You get:

{{{r=-1 +- 3sqrt(14)}}}.  Exclude the negative root, so Karen's rate was {{{-1 + 3sqrt(14)}}} mph, and Carla's was 2 mph faster or {{{1+3sqrt(14)}}} mph