Two poles, \( AB \) of length \( a \) meters and \( CD \) of length \( a+b \) (\( b\neq a \)) meters are erected at the same horizontal level with bases at \( B \) and \( D \). If \( BD=x \) and \( \tan \angle ACB=\frac{1}{2} \), then:
\( x^{2}+2(a+2b)x-b(a+b)=0 \)
\( x^{2}-2ax+a(a+b)=0 \)
\( x^{2}-2ax+b(a+b)=0 \)
\( x^{2}+2(a+2b)x+a(a+b)=0 \)
