From point A to point B two cars left at the same time. The first drove at a constant speed all the way.

From point A to point B two cars left at the same time. The first drove at a constant speed all the way. The second traveled the first half of the journey at a speed of 36 km / h, and the second half of the journey at a speed 54 km / h higher than the speed of the first, as a result of which he arrived at point B simultaneously with the first car. Find the speed of the first car.

Solution:
1. Let’s designate: x – speed of the first car, s – half way between points. Therefore, 2s is the total distance between points.
2. According to the condition of the problem, an equation was drawn up:
2s / x = s / 36 + s / (x + 54);
S is in each term, so it cancels out.
2 / x = 1/36 + 1 / (x + 54);
2 / x – 1 / (x + 54) – 1/36 = 0;
(2 * (x + 54) – x) / (x ^ 2 + 54x) – 1/36 = 0;
(36 * (2x + 108 – x) – (x ^ 2 + 54x)) / (36 * (x ^ 2 + 54x)) = 0;
(36 * (x + 108) – x ^ 2 – 54x) / (36 * (x ^ 2 + 54x)) = 0;
The fraction is 0 when the numerator is 0 and the denominator is not 0:
36 * (x + 108) – x ^ 2 – 54x = 0;
36x + 3888 – x ^ 2 – 54x = 0;
-x ^ 2 – 18x +3888 = 0;
Discriminant = (-18) ^ 2 + 4 * 1 * 3888 = 15876 (the root of 15876 is 126)
x = (18 + 126) / -2 or x = (18 – 126) / -2
x = -72 or x = 54
The speed cannot be negative, so it is 54 km / h.
Answer: The speed of the first car is 54 km / h.