<s>[INST] Alternate method for solving missing area question I recently saw a puzzle in an advert for the website Brilliant.org, which went as follows:

What is the blue area?
 
Hint: Think outside the box

My answer:
I set the area to be found to $x$, the side length of the square to be $y$, and the sections to be $a,b,c,d$ as below:

This then gave me the following equation to solve:
\begin{align}y^2&=2+3+4+x\\
&\Downarrow\\
x&=y^2-9\end{align}
And the following equations to do so:
\begin{align}\frac {ya}2 &=4\\
\frac {bc}2 &=3\\
\frac {yd}2 &=2\\
a+b&=y\\
c+d&=y\end{align}
I solved these to obtain:
$$a=2,
b=2,
c=3,
d=1,
y=4$$
And thus $$x=4^2-9=7$$
My question:
Is there another way I could have solved this, using the hint to think outside the box?
 [/INST] There is a wonderful generalization of the solution for this problem:
$$A = \sqrt{(a + b + c)^2 – 4ac}.$$
Where $a$, $b$ and $c$ are the known areas, and $A$ is the unknown area.
Caveat: $a$ and $c$ must be the triangles whose long legs are the side of the square, in this case $7$ and $9$.
$$A = \sqrt{729 – 4 \times 9 \times 7} = \sqrt{729- 252} = 21.84.$$
</s>