Solve {l}{5/3t_{1}=at_{2}}{t_{1}+t_{2}=12}{100=5/6t_1^2+1/2a}

$$\left. \begin{array} { l } { \frac { 5 } { 3 } t _ { 1 } = a t _ { 2 } } \\ { t _ { 1 } + t _ { 2 } = 12 } \\ { 100 = \frac { 5 } { 6 } t _ { 1 } ^ { 2 } + \frac { 1 } { 2 } a } \end{array} \right.$$

Solve for t_1, a, t_2

$$t_{1}=\frac{2\left(-13\sqrt{3}\cos(\frac{\arccos(\frac{1242\sqrt{3}}{2197})}{3})+6\right)}{3}\approx -10.976234382\text{, }a=\frac{5\left(-13\sqrt{3}\cos(\frac{\arccos(\frac{1242\sqrt{3}}{2197})}{3})+6\right)\left(13\sqrt{3}\cos(\frac{\arccos(\frac{1242\sqrt{3}}{2197})}{3})-12\right)}{9\left(169\left(\cos(\frac{\arccos(\frac{1242\sqrt{3}}{2197})}{3})\right)^{2}-48\right)}\approx -0.796202009\text{, }t_{2}=\frac{2\left(13\sqrt{3}\cos(\frac{\arccos(\frac{1242\sqrt{3}}{2197})}{3})+12\right)}{3}\approx 22.976234382$$

$$t_{1}=\frac{13\sqrt{3}\cos(\frac{\arccos(\frac{1242\sqrt{3}}{2197})}{3})+39\sin(\frac{\arccos(\frac{1242\sqrt{3}}{2197})}{3})+12}{3}\approx 12.373725911\text{, }a=\frac{5\left(13\sqrt{3}\cos(\frac{\arccos(\frac{1242\sqrt{3}}{2197})}{3})+39\sin(\frac{\arccos(\frac{1242\sqrt{3}}{2197})}{3})+12\right)}{3\left(-13\sqrt{3}\cos(\frac{\arccos(\frac{1242\sqrt{3}}{2197})}{3})-39\sin(\frac{\arccos(\frac{1242\sqrt{3}}{2197})}{3})+24\right)}\approx -55.181821531\text{, }t_{2}=\frac{-13\sqrt{3}\cos(\frac{\arccos(\frac{1242\sqrt{3}}{2197})}{3})-39\sin(\frac{\arccos(\frac{1242\sqrt{3}}{2197})}{3})+24}{3}\approx -0.373725911$$

$$t_{1}=\frac{13\sqrt{3}\cos(\frac{\arccos(\frac{1242\sqrt{3}}{2197})}{3})-39\sin(\frac{\arccos(\frac{1242\sqrt{3}}{2197})}{3})+12}{3}\approx 10.602508471\text{, }a=\frac{5\left(13\sqrt{3}\cos(\frac{\arccos(\frac{1242\sqrt{3}}{2197})}{3})-39\sin(\frac{\arccos(\frac{1242\sqrt{3}}{2197})}{3})+12\right)}{3\left(-13\sqrt{3}\cos(\frac{\arccos(\frac{1242\sqrt{3}}{2197})}{3})+39\sin(\frac{\arccos(\frac{1242\sqrt{3}}{2197})}{3})+24\right)}\approx 12.644690207\text{, }t_{2}=\frac{-13\sqrt{3}\cos(\frac{\arccos(\frac{1242\sqrt{3}}{2197})}{3})+39\sin(\frac{\arccos(\frac{1242\sqrt{3}}{2197})}{3})+24}{3}\approx 1.397491529$$