property of secants

Question

Find the value of x

Answer 1

The chords  from the intersection of the chords rule   AD(AD+DE)=AB(AB+BC)

AD=4, DE=8, AB=x, BC=2

substitute the values in AD(AD+DE)=AB(AB+BC)

4(4+8)=x(x+2)

16+32=x²+2x

48=x²+2x

Subtract 48 from each side.

48-48=x²+2x-48

x²+2x-48=0

x²+8x-6x-48=0                   (Factors on the left side)

x(x+8)-6(x+8)=0

(x+8) (x-6)=0                       (zero product property)

x+8=0   or  x-6=0

x=-8    or   x=6

The value of x is -8, 6.