Constant in a cdf with a jump

Let $X$ be a random variable with the distribution function $F(x) = \begin{cases} 0 & \text{if } x < 0 \ \alpha (1 + 2x^2) & \text{if } 0 \le x < 1 \ 1 & \text{if } x \ge 1 \end{cases}$, where $\alpha$ is a real constant. If the median of $X$ is $\frac{1}{\sqrt{2}}$, then the value of $\alpha$ equals