Tourists walked 6 km along a forest path, and then 10 km along a highway

Tourists walked 6 km along a forest path, and then 10 km along a highway, increasing their speed by 1 km / h. They spent 3.5 hours for the entire journey.

Let’s designate the speed of movement first as x km / h. Highway speed (x + 1) km / h.
Then we get the following equation:
6 / x + 10 / (x + 1) = 3.5
Let’s multiply both parts of the equation by the denominators:
x and 1 + x we ​​get:
x (10 / {x + 1} + 6 / {x}) = 7x / 2
{16 x + 6} {x + 1} = 7x / 2
16 x + 6 = 7 x ^ {2} / 2 + 7 x / 2
Move the right side of the equation to the left side of the equation with a minus sign.
The equation becomes
– 7 x ^ {2} / {2} + 25 x / 2 + 6 = 0
This is an equation of the form
a * x ^ 2 + b * x + c = 0
The quadratic equation can be solved using the discriminant.
where D = b ^ 2 – 4 * a * c
Because
a = -7/2
b = 25/2
c = 6
, then
D = b ^ 2 – 4 * a * c = (25/2) ^ 2 – 4 * (-7/2) * (6) = 961/4
Because D> 0, then the equation has two roots.
x1 = (-b + sqrt (D)) / (2 * a)
x2 = (-b – sqrt (D)) / (2 * a)
or
x_ {1} = – 3/7 (matches the conditions)
x_ {2} = 4
Let’s get the answer: first, the tourists walked at a speed of 4 km / h, then at a speed of 5 km / h.