Parabolic geometry!

The two roots of the equation $x^2 - bx + c$ are: $x_{1} = \dfrac{-b-\sqrt{b^2 - 4c}}{2} \quad \quad \quad x_{2} = \dfrac{-b+\sqrt{b^2 - 4c}}{2}$

$| x_{1} - x_{2} | = \cancel{\dfrac{-b}{2} - \dfrac{(-b)}{2}} + \dfrac{\sqrt{b^2 - 4c}}{2} - \bigg(-\dfrac{\sqrt{b^2 - 4c}}{2}\bigg)$

$| x_{1} - x_{2} | = \sqrt{b^2 - 4c}$

$\begin{aligned} {\color{#EC7300}{\text{Area}}} &= \bigg(| x_{1} - x_{2} |\bigg) \\ &= \bigg(\sqrt{b^2 - 4c}\bigg)^2 \\ &= b^2 - 4c \\ &= \boxed{\color{#EC7300}{15}} \end{aligned}$

From the quadratic formula, the roots of the polynomial $f(x) = x^2 - bx + c$ are found at

$x = \frac{-b \pm \sqrt{b^2 - 4c}}{2}$ ,

Thus, the difference between the roots is $\frac{-b + \sqrt{b^2 - 4c}}{2} - \frac{-b - \sqrt{b^2 - 4c}}{2} = \sqrt{b^2 - 4c}$

This difference is the length of the base of the orange square, thus the area of the square is $b^2 - 4c = 15$

Area of the orange square=Square of difference of roots $=b^{2}-4c=15$

See here to get the above formula