Maths question?

Let y cm be the length of BP.
Then, the length of AP = (y + 8) cm

AB × AP = BP²
8 (y + 8) = y²
y² - 8y - 64 = 0
y = [8 ± √(8² + 4×64)]/2
y = 4 + 4√5  or  y = 4 - 4√5 (rejected)
y + 8 = 12 + 4√5

The length of AP = 12 + 4√5 cm ≈ 20.94 cm
The length of BP = 4 + 4√5 cm = 12.94 cm

The interval AB is extended to the point P so that AB x AP = BP^2 
 o------------------o-------------o

 A                      B               P

 If AB = 8 cm 
 8*(8 + x) = x^2
 x^2 - 8x - 64 = 0
 (x - 4)^2 - 80 = 0
 x = 4 (1 + sqrt(5))
 The length of AP = 4 (2 + sqrt(5))
 and the length of BP = 4 (1 + sqrt(5)).

let AB=a;BP=b

a*(a+b)=b^2

a^2+ab-b^2 =0

a^2+ab-b^2+2b^2-2b^2+ab-ab=0

a^2+2ab+b^2-2b^2-ab=0

(a+b)^2-(b^2+ab)

64=(b^2+ab)

a*(a+b)=b^2 given

a^2=b=64

a=8;b=64

AB = 8
BP = x
AP = AB + BP = 8 + x

AB * AP = BP^2
8 * (8 + x) = x^2
64 + 8x = x^2
64 = x^2 - 8x
64 = x^2 - 2 * 4x
64 = x^2 - 2 * 4x + 4^2 - 4^2
64 + 4^2 = x^2 - 2 * 4x + 4^2
64 + 16 = (x - 4)^2
80 = (x - 4)^2
16 * 5 = (x - 4)^2
+/- 4 * sqrt(5) = x - 4
x = 4 +/- 4 * sqrt(5)

x > 0

x = 4 + 4 * sqrt(5)
x = 4 * (1 + sqrt(5))

BP = 4 * (1 + sqrt(5)) cm

4 * (1 + sqrt(5)) =>
4 * (1 + 2.236) =>
4 * 3.236 =>
4 * 3.2 + 4 * 0.036 =>
12.8 + 0.144 =>
12.944

BP is roughly 12.944 cm in length