The best method to solve the equation is employing the Quadratic formula method
which is,
[tex]x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}[/tex]
The equation given is,
[tex]3t^2-5t+7=-3[/tex]
Add both sides by 3
[tex]\begin{gathered} 3t^2-5t+7+3=-3+3 \\ 3t^2-5t+10=0 \end{gathered}[/tex]
Then, solve with the quadratic formula
[tex]t_{1,\: 2}=\frac{-\left(-5\right)\pm\sqrt{\left(-5\right)^2-4\cdot\:3\cdot\:10}}{2\cdot\:3}[/tex]
Simplify the formula above
[tex]t_{1,2}=\frac{5\pm\sqrt[]{25-120}}{6}=\frac{5\pm\sqrt[]{-95}}{6}[/tex]
Note that
[tex]\sqrt[]{-95}=\sqrt[]{-1}\times\sqrt[]{95}=\sqrt[]{95}i[/tex]
Therefore,
[tex]t_{1,\: 2}=\frac{5\pm\sqrt{95}i}{6}[/tex]
Separate the solution
[tex]t_1=\frac{5+\sqrt{95}i}{6},\: t_2=\frac{5-\sqrt{95}i}{6}[/tex]
Rewrite the solution in standard complex form
[tex]t_1=\frac{5}{6}+\frac{\sqrt{95}}{6}i,t_2=\frac{5}{6}-\frac{\sqrt{95}}{6}i[/tex]
Hence, the solutions to the quadratic equation are
[tex]t_1=\frac{5}{6}+i\frac{\sqrt[]{95}}{6},t_2=\frac{5}{6}-i\frac{\sqrt[]{95}}{6}[/tex]
