Solve (12-lambda)(4t^2-lambda)-36t^2=0

$$( 12 - \lambda ) ( 4 t ^ { 2 } - \lambda ) - 36 t ^ { 2 } = 0$$

Solve for t (complex solution)

$$t=-\frac{i\left(3-\lambda \right)^{-\frac{1}{2}}\sqrt{\lambda }\sqrt{\lambda -12}}{2}$$

$$t=\frac{i\left(3-\lambda \right)^{-\frac{1}{2}}\sqrt{\lambda }\sqrt{\lambda -12}}{2}\text{, }\lambda \neq 3$$

Solve for t

$$t=\frac{\sqrt{\frac{\lambda \left(12-\lambda \right)}{3-\lambda }}}{2}$$

$$t=-\frac{\sqrt{\frac{\lambda \left(12-\lambda \right)}{3-\lambda }}}{2}\text{, }\left(\lambda \geq 0\text{ and }\lambda <3\right)\text{ or }\lambda \geq 12$$

Solve for λ

$$\lambda =2\left(-\sqrt{t^{4}+3t^{2}+9}+t^{2}+3\right)$$

$$\lambda =2\left(\sqrt{t^{4}+3t^{2}+9}+t^{2}+3\right)$$