1. The problem is to solve for $t$ in the displacement equation $$d = v_i t + 0.5 a t^2$$ where $d$ is displacement, $v_i$ is initial velocity, $a$ is acceleration, and $t$ is time. 2. This is a quadratic equation in terms of $t$. The general quadratic form is $$at^2 + bt + c = 0$$ where here, $$a = 0.5 a, \quad b = v_i, \quad c = -d$$. 3. To solve for $t$, use the quadratic formula: $$t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 4. Substitute the values: $$t = \frac{-v_i \pm \sqrt{v_i^2 - 4 \times 0.5 a \times (-d)}}{2 \times 0.5 a}$$ 5. Simplify inside the square root and denominator: $$t = \frac{-v_i \pm \sqrt{v_i^2 + 2 a d}}{a}$$ 6. Therefore, the two possible solutions for $t$ are: $$t = \frac{-v_i + \sqrt{v_i^2 + 2 a d}}{a} \quad \text{or} \quad t = \frac{-v_i - \sqrt{v_i^2 + 2 a d}}{a}$$ 7. In physical problems, choose the solution for $t$ that is positive and makes sense in context.