At some height above the surface of the earth, the projectile exploded into several fragments that flew

At some height above the surface of the earth, the projectile exploded into several fragments that flew in all directions at the same speeds. The splinter, flying vertically downward, reached the ground in time t₁. A splinter flying vertically upward fell to the ground after time t₂. How long did the fragments fall, flying horizontally? Disregard air resistance.

decision
H = v * t1 + g * t1 squared / 2
h = – v * t + g * t 2 squared / 2

v = h / t1 – g * t1 / 2
v = – h / t2 + g * t2 / 2

h / t1 – g * t1 / 2 = -h / t2 + g * t 2/2
h / t1 + h / t2 = g * t1 / 2 + g * t 2/2
h * (t1 + t2) / (t1 * t2) = g * (t1 + t2) / 2
h = g * t1 * t2 / 2

for horizontal fragments
h = g * t squared / 2
h = g * t1 * t2 / 2 = g * t squared / 2
g * t1 * t2 / 2 = g * t squared / 2
t1 * t2 = squared
got t = root of (t1 * t2)
answer t = root of (t1 * t2)