Sum and Product of Roots

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The quadratic formula
 

$ x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a} $

 

give the roots of a quadratic equation which may be real or imaginary. The ± sign in the radical indicates that
 

$ x_1 = \dfrac{-b + \sqrt{b^2-4ac}}{2a} $   and   $ x_2 = \dfrac{-b - \sqrt{b^2-4ac}}{2a} $

 

where x1 and x2 are the roots of the quadratic equation ax2 + bx + c = 0. The sum of roots x1 + x2 and the product of roots x1·x2 are common to problems involving quadratic equation.
 

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} $

 

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