1. **State the problem:** We need to make $t$ the subject of the formula given by $$S = ut + \frac{1}{2}at^2$$ 2. **Identify the formula and what it represents:** This is the equation of motion for displacement $S$ under constant acceleration $a$, initial velocity $u$, and time $t$. 3. **Rewrite the equation:** $$S = ut + \frac{1}{2}at^2$$ 4. **Rearrange the equation to standard quadratic form in $t$:** $$\frac{1}{2}at^2 + ut - S = 0$$ 5. **Use the quadratic formula to solve for $t$:** The quadratic formula for $ax^2 + bx + c = 0$ is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Here, $a = \frac{1}{2}a$, $b = u$, and $c = -S$. 6. **Substitute into the quadratic formula:** $$t = \frac{-u \pm \sqrt{u^2 - 4 \times \frac{1}{2}a \times (-S)}}{2 \times \frac{1}{2}a}$$ 7. **Simplify inside the square root:** $$t = \frac{-u \pm \sqrt{u^2 + 2aS}}{a}$$ 8. **Final expression for $t$:** $$\boxed{t = \frac{-u \pm \sqrt{u^2 + 2aS}}{a}}$$ This gives two possible values for $t$, depending on the sign chosen. In physical contexts, choose the value that makes sense (usually positive time).