maths question?

Let s km/h be the speed of the car.

Time taken = Distance/Speed
1/(s - 10) - (1/s) = 1

For s ≠ 0 and s - 10 ≠ 0
[500/(s - 10) - (500/s)] s(s - 10) = s(s - 10)
500s - 500(s - 10) = s² - 10s
500s - 500s + 5000 = s² - 10s
s² - 10s - 5000 = 0
s = [10 ± √(10² + 4×5000)]/2
s = 5 + 5√201  or  s = 5 - 5√201 (rejected)
s ≈ 75.89

Speed of the car = 5 + 5√201 km/hr ≈ 75.89 km/hr

Let v km/h be that constant speed, then
500/(v-10)-500/v=1
=>
500[v-(v-10)]=v(v-10)
=>
v^2-10v-5000=0
=>
v=75.88724 or -65.88724 (rejected)
=>
v=75.89 km/h approximately.

You are given that:

v = 500/t  & v-10 = 500/(t+1). Convince yourself of these.

Now, just solve this system. U know how to do that?

distance = rate * time

The distance was 500 km at a fixed speed:

d = rt
500 = rt

If he was 10 km/h less it would add an hour.  Subtracting 10 to the rate gives us (r - 10) and adding one to the time gives us (t + 1).  The distance is still the same so now we have this equation:

d = rt
500 = (r - 10)(t + 1)

Expanding that to get:

500 = rt + r - 10t - 10

We now have a system of two equations and two unknowns.  Since we only need the rate of the car (r), solve the first equation for t in term of r, then substitute into the other equation:

500 = rt
500 / r = t

500 = rt + r - 10t - 10
500 = r(500 / r) + r - 10(500 / r) - 10
500 = 500 + r - 5000 / r - 10
0 = r - 5000 / r - 10


Now let's multiply both sides by r to make a quadaratic:

0 = r² - 5000 - 10r
0 = r² - 10r - 5000

Solve for r:

r = [ -b ± √(b² - 4ac)] / (2a)
r = [ -(-10) ± √((-10)² - 4(1)(-5000))] / (2 * 1)
r = [ 10 ± √(100 + 20000)] / 2
r = [ 10 ± √(20100)] / 2
r = [ 10 ± √(100 * 201)] / 2
r = (10 ± 10√201) / 2
r = 5 ± 5√201

We can't have an initial rate that is negative, so throwing that out we get:

r = 5 + 5√201 km/h

as the starting velocity.

v = d/t = 500/t, so t = 500/v
v - 10 = 500/(t+1)
(v - 1)(t + 1) = 500
vt + v - t - 1 - 500 = 0
v(500/v) + v - (500/v) - 501 = 0
v - 1 - (500/v) = 0
v^2 - v - 500 = 0
v = (-(-1) +/- sqrt((-1)^2 - 4(1)(-500))) / (2*1)
v = (1 +/- sqrt(1 + 2000)) / 2
v = (1 +/- sqrt(2001)) / 2
v = 0.5 +/- sqrt(500.25)
v =~ 22.866269246345041707987157489031 or -21.866269246345041707987157489031
But a negative number doesn't make sense, so therefore:
v = 0.5 + sqrt(500.25) =~ 22.866269246345041707987157489031

The speed is 75.8872 km/hr.  Try it and see!