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 is known as a quadratic equation and the formula for its roots is
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
Fact 2: product of roots is
Fact 3: difference of squares i. e.
Let's introduce 2 new variables:
1) difference between the 2 roots . i . e
2) mean of the 2 roots . i . e
In the graph below, notice that and are the 2 roots) because and .
So where p is the product of the 2 roots.
Hence, the 2 roots are
The formula for the roots of a quadratic equation 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
We have and
Hence, by our above formula
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