To solve the equation \( t - 122 = 3t^2 - 3 \), we first rearrange the equation to move all terms to one side:
\[ 3t^2 - t - 119 = 0 \]
Next, we use the quadratic formula to solve for \( t \), which is given by:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our equation \( 3t^2 - t - 119 = 0 \), the coefficients are:
- \( a = 3 \)
- \( b = -1 \)
- \( c = -119 \)
Plugging these values into the quadratic formula:
- Calculate the discriminant \( b^2 - 4ac \):
\[ (-1)^2 - 4 \cdot 3 \cdot (-119) = 1 + 1428 = 1429 \]
- Now we can use the quadratic formula:
\[ t = \frac{-(-1) \pm \sqrt{1429}}{2 \cdot 3} = \frac{1 \pm \sqrt{1429}}{6} \]
Since \( \sqrt{1429} \) is not a perfect square, we need to approximate its value to find possible \( t \) values:
\[ \sqrt{1429} \approx 37.8 \]
Thus, we can calculate \( t \):
\[ t \approx \frac{1 + 37.8}{6} \approx \frac{38.8}{6} \approx 6.47 \]
and
\[ t \approx \frac{1 - 37.8}{6} \approx \frac{-36.8}{6} \approx -6.13 \]
Since neither of these values matches the options provided (−3, −1, 1, 3), we should check the options directly or verify the discriminant. The values in the responses do not seem derived from the roots calculated as above and indicate potential mistakes or lack of proper rounding or roots checking.
However, since the equation \( 3t^2 - t - 119 = 0 \) was derived correctly, we accept that there may not be a solution among the options given.
Please verify the correct options or see if \( t = 3 \) or other values were misinterpreted or miscalculated. However, based on direct evaluation, those values from responses do not resolve solely from the quadratic calculated. Consider re-evaluating each proposed root directly in the full original equation if matching needs depth verification, leading to 'none of the above' or correctness validated from calculated roots.
If we need clarification on which to select from potential responses, based on negative vs positive checks in values needing check placement, ensuring originally derived coefficients hold on quick sequential checks.
Conclusion:
Therefore, no proposed responses match the computed values for \( t \). The investigation shows \( t = \frac{1 + \sqrt{1429}}{6} \) and \( t = \frac{1 - \sqrt{1429}}{6} \), with numerical estimates leading outside options provided.