Okay, so we have a polynomial equation of the form ax^2+bx+c = 0 and want to solve for x.
To do that, we need to get x alone.
First, we get all the terms without an x to the other size of the equation:
ax^2 + bx = -c
Then we divide by a since it makes the math less messy
x^2 + bx/a = -c/a
Alright, now we want to get the left hand side into a form where there is only a single x so that we can actually solve for it.
x^2 + bx/a is equal to (x+d)^2-e for some constants d and e.
We want the form on the right hand side since we're able to isolate x when we set it to 0.
Let's interpret geometrically.
We can interpret x^2 + bx/a geometrically as a square of side length x, and a rectangle of side lengths x and b/a.
If we split up the b/a into two evenly sized rectangles and place them flush with the square, we are able to get close to what we want.
These rectangles have side lengths x and b/2a.
We want the total area to be a square, so we just need to find the missing square in the lower right.
The area of the missing square is just the square of the rectangle's non-x side length. aka (b/2a)^2 = b^2/(4a^2)
Since the whole figure is a square of a known area, we can say (x + b/(2a))^2 = x^2 + bx/a + b^2/(4a^2)
Subtract b^2/(4a^2) from both sides and the right hand side is identical to our earlier equation:
(x + b/(2a))^2 - b^2/(4a^2) = x^2 + bx/a = -c/a
Simplify and solve for x:
(x + b/(2a))^2 - b^2/(4a^2) = -c/a
(x + b/(2a))^2 = -c/a + b^2/(4a^2)
(This is the equation your pic stops at, just get the -c/a to have the same denominator and it looks identical)
x + b/(2a) = sqrt(-c/a + b^2/(4a^2))
x = sqrt(-c/a + b^2/(4a^2)) - b/2a
This can be rewritten in the standard quadratic formula form if you feel like doing a bit of algebraic rearrangement.
