A and B start from the same point and travel west and north, respectively. A travels 3 miles per hour faster than B. At the end of two hours, they are 30 miles apart. Find their distances.

I believe you're actually looking for their respective rates. If A travels directly west and B travels directly north, and the distance between them is 30 miles, what we have is a right triangle situation. The hypotenuse of the triangle is 30. If the distance an object travels at a certain rate for given time is d = rt, then for B, our formula for distance is d = 2r (since the time each traveled is 2 hours). A traveled 3 miles per hour faster, so the formula for distance for A is d = (r + 3)2. Again, each traveled for 2 hours, so t = 2. Distributing we get that d = 2r + 6. Now that we have each expression for A and B, we use them in Pythagorean's Theorem to find the only unknown we have which is r. That's why I said in the beginning that I believe what you're actually looking for is the rate that each traveled. Our equation is:

[tex]30^2=(2r+6)^2+(2r)^2[/tex] and

[tex]900=4r^2+24r+36+4r^2[/tex] and

[tex]900=8r^2+24r+36[/tex]

Putting everything on one side and setting the quadratic equal to 0 to factor, we get

[tex]8r^2+24r-864=0[/tex]

You can factor out an 8 to make the numbers a bit smaller:

[tex]8(r^2+3r-108)=0[/tex]

Factor to get

8(r - 9)(r + 12)=0

That means, by the Zero Product Property, that 8 = 0, r - 9 = 0, or r + 12 = 0. We all know that 8 doesn't = 0, so forget that one!! If r - 9 = 0, then r = 9. If r + 12 = 0. then r = -12. We all know that rate cannot be negative (velocity can, but we are not using vector math here), so we discount rate as -12. That means that r = 9. B's rate is 9 then and A's rate is 12. There you go!