@rabotaakk-nw9nm

5:35   AE=4+2b-4=2b=20/3
ΔAPE~ΔAQD => AQ/AD=AP/AE
x/8=4/(20/3); x=8×4×3/20=24/5

@jimlocke9320

We note that tan(α) = OC/AC = 2/4 = 1/2. Applying the tangent double angle formula, tan(2α) = (2tan(α))/(1 - tan²(α)) = (2)(1/2)/(1 - (1/2)²) = 4/3. Using the designations for points given in the video at 1:35 and constructing line segments as needed, PE/AP = tan(2α) so PE/4 = 4/3 and PE = 16/3. So, by Pythagoras, AE = √(4)² + (16/3)²) = √((144/9) + (256/9)) = √(400/9) = 20/3. ΔAPE and ΔAQD are similar, so AD/AE = AQ/AP, 8/(20/3) = X/4 and X = 24/5 units, as GrayYeon Math also found.

@soli9mana-soli4953

Tracing the perpendicular radius CH to line AQ we know that radius splits the chord in two equal parts, so we can calculate AH  as:
AH = AC*cos 2 alpha
knowing that:
cos alpha = AC/AO = 4/2√ 5   then cos 2 alpha = 3/5  
and  AH = X/2,  AC =4
x/2 = 4*3/5 
X = 24/5