I have several questions regarding the following solution,

 1. How did the author of the solution know that the parabola passing through the center of the axes?

 2. How was the area under the half of the parabola and the x axis calculated? 

____

> We draw two points $p$ and $q$ at random from the interval $[−1,
 1]$. 
> 
> Let $x_1$ and $x_2$ denote the roots of the equation $x^2 + px + q =
 0$. 
> 
> Find the probability that
> 
> (a) $x_1, x_2 \in \mathbb{R}$,
> 
> (b) $x_1 + x_2 < 1$.


Given that, $x^2 + px + q = 0$, $\Delta = p^2-4q$.

**(a)**

[![enter image description here][1]][1]

$x_1, x_2 \in \Bbb R \Rightarrow \Delta \ge 0$

$\Omega = [-1, 1]^2 = \{(p,q) \in \Bbb R^2: p,q \in [-1,1]\}$ 

$A$ = roots exist.

$A = \{(p,q) \in [-1,1]^2 : p^2 -4q \ge 0 \}$

$x^2 - 4y \ge 0$

$y \le \frac{x^2}{4}$

$\Bbb P(A) = \frac{area-under-the-parabola}{4} = \frac{2 + 2 \cdot  half-of-parabola }{4}$

$= \frac{1}{2} \cdot \frac{1}{2} \cdot \int_{0}^{1}\frac{x^2}{4}\, dx$ 

$ = \frac{13}{24}$


  [1]: https://i.sstatic.net/ZjK93.png