Find the point where locus crosses the damping ratio line?

Solve the closed-loop equation: $$T(s) = frac{G(s)}{1+G(s)} = frac{K(s^2-6s+8)}{s^2+6s+25+K(s^2-6s+8)}$$

structure the denominator to the following: $$s^2 + as+b$$ and relate to the denominator of the standard second order mass model: $$H(s) = frac{Qomega^2}{s^2+2zetaomega s+ omega^2}$$ where $Q$ is any arbitrary gain. Relate $K$ to $zeta$ such that $zeta = 0.5$. $$T(s) = frac{K(s^2-6s+8)}{s^2(1+K)+s(6-6K)+25+8K}$$ $$T(s) = frac{K(s^2-6s+8)/(1+K)}{s^2+s(6-6K)/(1+K) + (25+8K)/(1+K)}$$ $$2zeta = frac{6-6K}{1+K}cdotfrac{sqrt{1+K}}{sqrt{25+8K}}$$ $$2zeta = frac{(6-6K)sqrt{1+K}}{sqrt{1+K}^2sqrt{25+8K}} = frac{(6-6K)}{sqrt{(1+K)(25+8K)}} = 1$$ $$rightarrow 8K^2+33K+25 = (6-6K)^2$$ Solve this and mind the sign. due to the squared structure, on $K$ value will yield a damping value of $-0.5$. so plug the result back into the equation to verify the correct one.