How to Solve Quadratic Equations Without using Quadratic Formula?

We are very much aware of the quadratic formula to solve quadratic equations. Have you ever wondered about solving a quadratic equation without using quadratic formula?
Presenting here one very elegant approach to solve the quadratic equation without using the quadratic formula (of course not using factorization indeed 😊)
 Starting from S, imagine a robot, facing east, moving according to these instructions:



Step (i). Move forward by an amount equal to the coefficient of x^2 (backward if negative)
Step (ii). Rotate anti-clockwise
Step (iii). Move forward (or backward) by an amount equal to the coefficient of x.
Step (iv). Now Rotate anti-clockwise
Step (v).  Move forward (or backward) to the terminal point T by an amount equal to the constant term.

Red and blue lines are then drawn using a set-square from S to the middle line (or its extension) and on to T. (Note in case Q2, the absence of the coefficient of x causes a double rotation).

The roots are given by the tangent of the two angles formed in a clockwise sense from the first horizontal to the line of the first move. (Only the red angle is shown).

 Note: The method was first introduced by Edward Lill (1830-1900) an Austrian engineer who turned into an army officer.

Let us see one Example now: 

Here is an illustration for searching for roots of the equation


(roots  are and ) by Lill's method, look at the animation below.





Happy Math Learning!