The third proportional to $ x $ and $ x + 100 $ is 405, find the value of ( where$ x $ >100)

The third proportional to $x$ and $x+100$ is 405.
So we use:
$x : (x+100) = (x+100) : 405$
$ x \cdot 405 = (x+100)^2$
$405x = x^2 + 200x + 10000$
$x^2 - 205x + 10000 = 0$
Solve quadratic:
$x = \dfrac{205 \pm \sqrt{205^2 - 4 \cdot 10000}}{2}$
$205^2 = 42025$
$\sqrt{42025 - 40000} = \sqrt{2025} = 45$
So,
$x = \dfrac{205 \pm 45}{2}$
Possible values:
$x = \dfrac{250}{2} = 125$
$x = \dfrac{160}{2} = 80$

Given $x > 100$, so the valid answer is:

$x = 125$