How do I do this: Using the quadratic formula, derive the formulas for sum and products.?

Question

THIS IS THE QUADRATIC FORMULA:
http://media.nasaexplores.com/lessons/01-003/images/quadform.gif

and this is formula for the sum of the roots: (-b/a) and the formula for the product of the roots is: (c/a)

How do I derive thee formulas for sum and product from the quadratic formula?

THANKS!

Answer 1

The quadratic formula gives two roots, let's call them
x1 = [-b + sqrt(b^2-4ac)]/(2a)
and
x2 = [-b - sqrt(b^2-4ac)]/(2a)

If we add them we get the sum of the roots
x1 + x2 = -b/(2a) - b/(2a) (because the + sqrt and - sqrt terms cancel)

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

Similarly, the product of the roots is x1*x2 =
[-b*-b - (b^2-4ac)]/(2a)
(because the terms bsqrt(b^2-4ac) from doing the multiplication are of opposite sign and cancel, also sqrt(b^2-4ac)^2 = b^2-4ac)

= 4ac/(2a)
=c/a.

Answer 2

x= [-b + sqr( b^2 -4ac ) ]/2a

=(1/2) [ (-b/a) + sqr( (-b/a)^2 -4(c/a) ) ]
= [ sum +/- sqr( sum^2 -4 product) ]/2

Answer 3

r1 = -b/(2a) + sqrt(b^2-4ac)/(2a)
r2 = [-b/(2a) - sqrt(b^2-4ac)/(2a)
r1+r2 =-b/2a -b/2a = -2b/2a = -b/a

r1r2 = b^2/4a^2 -(b^2-4ac)/4a^2
= (b^2-b^2 +4ac)/ (4a^2)
= 4ac/(4a^2) = c/a