Answer
55 mph and 40 mph
Work Step by Step
For 220 miles, speed of $x$
For 80 miles, speed of $x-15$
Total time was 6 hours
$300 = x*(220/x)+(x-15)*(80/(x-15))$
$220/x + (80/(x-15)) = 6$
$220/x + (80/(x-15)) = 6$
$220/x*(x)(x-15) + (80/(x-15))*(x)(x-15) = 6*(x)(x-15)$
$220*(x-15)+80*x = 6*(x^2-15x)$
$220x-3300+80x=6x^2-90x$
$300x-3300=6x^2-90x$
$300x-3300-300x+3300=6x^2-90x-300x+3300$
$0 = 6x^2-390x+3300$
$0= (6x^2-390x+3300)/6$
$0= x^2-65x+550$
$0=(x-55)(x-10)$
$x-55=0$
$x=55$
$x-10=0$
$x=10$
$x\ne 10$ since the last 80 miles had the speed reduced by 15 mph.
$x=55$
$x-15$
$55-15=40$
