Question

find the value of ( x ). note that ( ab = 2 ), ( bc = x ), and ( cd = 3 ).

Explanation:

Step1: Apply secant-tangent theorem

For a tangent segment \(AB\) and a secant segment \(BD\) intersecting the circle at \(C\) and \(D\), the theorem states \(AB^2 = BC \times BD\). First, express \(BD\) as \(BC + CD = x + 3\).

Step2: Substitute known values

Substitute \(AB=2\), \(BC=x\), \(BD=x+3\) into the formula:
\(2^2 = x(x + 3)\)

Step3: Simplify and solve quadratic

Expand and rearrange into standard quadratic form:
\(4 = x^2 + 3x\)
\(x^2 + 3x - 4 = 0\)
Factor the quadratic:
\((x + 4)(x - 1) = 0\)
Since length cannot be negative, discard \(x=-4\).

Answer:

\(x=1\)