Monday, January 14, 2013


Derivation of the Sum of Roots
$ x_1 + x_2 = \dfrac{-b + \sqrt{b^2-4ac}}{2a} + \dfrac{-b - \sqrt{b^2-4ac}}{2a} $
$ x_1 + x_2 = \dfrac{-b + \sqrt{b^2-4ac} - b - \sqrt{b^2-4ac} }{2a} $
$ x_1 + x_2 = \dfrac{-2b}{2a} $
 
$ x_1 + x_2 = - \, \dfrac{b}{a} $

Derivation of the Product of Roots
$ x_1 \, x_2 = \left( \dfrac{-b + \sqrt{b^2-4ac}}{2a} \right) \left( \dfrac{-b - \sqrt{b^2-4ac}}{2a} \right) $
By difference of two squares:
$ x_1 \, x_2 = \dfrac{b^2 - (b^2-4ac)}{4a^2} $
$ x_1 \, x_2 = \dfrac{4ac}{4a^2} $
$ x_1 \, x_2 = \dfrac{c}{a} $

The formulas

sum of roots: −b/a
product of roots: c/a 
As you can see from the derivation below, when you are trying to solve aquadratic equations in the form of ax2+bx +c. The sum and product of the roots can be rewritten using the two formulas above.Sum and Product Formula Derived
Example 1
The example below illustrates how this formula applies to the quadratic equation x2 + 5x +6. As you can see the sum of the roots is indeed -b/a and the product of the roots is c/a.

Picture of sum and product of roots formula
Example 2
The example below illustrates how this formula applies to the quadratic equation x2 - 2x - 8. Again, both formulas--for the sum and the product boil down to -b/a and c/a, respectively.

Picture of sum and product of roots formula
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1 comment:

  1. UnknownNovember 24, 2020 at 1:11 AM

    You can visit https://quadraticformulacalculator.net/ to access free calculators.

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