Kinematics from Lagrangian

Plugging the given Lagrangian into the Euler-Lagrange's equation gives the equation of motion: $\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}}\right)=\frac{\partial L}{\partial x} \implies \ddot{x} = 0.5$ $x(0) = 0 \ ; \ \dot{x}(0) = 1$ The solution to this equation of motion is: $x(t) = \frac{t^2}{4}+t$ To find the time at which $x=5$ , the following quadratic equation is to be solved: $t^2 + 4t - 20=0 \implies t = \frac{-4 \pm \sqrt{96}}{2}$ The positive root of this equation is the required answer which is: $\boxed{ t \approx 2.899}$