A Better Way to Solve a Quadratic Equation

Hello everyone. Here is an alternate(somewhat simpler) method of solving a quadratic equation. I learnt it from a youtube stream by "3 Blue 1 Brown". The method was originally given by Po Shen Loh.

The Method is Below:

We all know that the equation of the form $ax^{2}+bx+c=0$ is known as a quadratic equation and the formula for its roots is $\frac{-b \pm \sqrt{b^{2}-4ac}}{2a}$.
But there is a shorter formula which is much easier to use while solving problems. You just have to know the three facts below and then after a little manipulation, you are done!

Fact 1: sum of roots is $\frac{-b}{a}$
Fact 2: product of roots is $\frac{c}{a}$
Fact 3: difference of squares i. e. $(m-d)(m+d)=m^{2}-d^{2}$

Let's introduce 2 new variables:
1) $d =$ difference between the 2 roots . i . e $|a-b|$
2) $m =$ mean of the 2 roots . i . e $\frac{a+b}{2}$
In the graph below, notice that $\alpha\beta= (m+d)(m-d) = m^{2}-d^{2}=\frac{c}{a}$ $(\alpha$ and $\beta$ are the 2 roots) because $\alpha = m+d$ and $\beta = m-d$.
So $d^{2}= m - p$ where p is the product of the 2 roots.
Hence, the 2 roots are $m \pm \sqrt{m^{2}-p}$

The formula for the roots of a quadratic equation $m \pm \sqrt{m^{2}-p}$ where p is the product of roots and m is the midpoint of the roots which can be derived from the 3 above facts.

Now, I will do an example of this: Solve the equation $x^{2}+3x+1=0$

We have $m=\frac{-3}{2}$ and $p=\frac{1}{2}$
Hence, by our above formula $x = \frac{-3}{2} \pm \frac{\sqrt{5}}{2}$

The traditional quadratic formula can be derived from this and vice versa. If you have any questions or suggestions, please ask them in the comments. Try more examples to get the hang of it. Hope you liked this note and will start using this formula in the future!

Diagram is below