<PRE><B>Question 1102120</B>
&lt;br&gt;
The fastest way to solve this problem is by trial and error, hoping that the speeds are whole numbers.&lt;br&gt;
{{{258 = 2*129 = 2*3*43}}}
So a good guess for the first part of the trip is 6 hours at 43 mph.&lt;br&gt;
That would leave 4 hours for the rest of the trip, which was 208 miles; 208 miles in 4 hours means 52 mph; and indeed 52mph is 9 mph faster than 43 mph.&lt;br&gt;
Algebraically...&lt;br&gt;
Let x and x+9 be the two speeds.  Then 258 miles at speed x plus 208 miles at speed x+9 makes a total of 10 hours:&lt;br&gt;
{{{258/x + 208/(x+9) = 10}}}
{{{258(x+9) + 208x = 10x(x+9)}}}
{{{258x + 2322 + 208x = 10x^2+90x}}}
{{{10x^2 - 370x - 2580 = 0}}}
{{{x^2 - 37x - 258 = 0}}}
{{{(x-43)(x+6) = 0}}}&lt;br&gt;
The lower speed is 43 mph; the higher speed is 52 mph.&lt;br&gt;
Note that in the algebraic solution, we had to factor the quadratic x^2-37x-258; to do that, we had to find two numbers whose product was 258.&lt;br&gt;
But that&#39;s exactly what we did in the first place.  So the algebraic solution didn&#39;t make the work any easier; it only made us do more work (a LOT more!) to get to the answer. </PRE>