Saturday, May 14, 2022

The Sum and Product of Roots of a Quadratic Equation

The Sum and Product of Roots of a Quadratic Equation


Introduction


Let s, t be the roots of the equation a*x^2 + b*x + c = 0.


Let:


s = (-b + √(b^2 - 4*a*c)) / (2 * a)

t = (-b - √(b^2 - 4*a*c)) / (2 * a)


Then


s + t = -b / a

s * t = c / a


We see this topic a lot in algebra, let's see how these properties are derived. Fairly simple.  


Sum of the Roots


s + t

=  (-b + √(b^2 - 4*a*c)) / (2 * a) + (-b - √(b^2 - 4*a*c)) / (2 * a)

= (-2 * b) / (2 * a)

= -b / a


Product of the Roots


s * t 

=  (-b + √(b^2 - 4*a*c)) / (2 * a) * (-b - √(b^2 - 4*a*c)) / (2 * a)

= (b^2 + b * √(b^2 - 4*a*c) - b * √(b^2 - 4*a*c) - (b^2 - 4*a*c)) / (4*a^2)

= (b^2 - b^2 + 4*a*c) / (4*a^2)

= (4*a*c) / (4*a^2)



Eddie



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