Hi apdshaft!!
The quadratic equation a.x^2 + b.x + c = 0
has the following solutions:
x1 = (- b + sqrt(b^2 - 4.a.c)) / 2.a
and
x2 = (- b - sqrt(b^2 - 4.a.c)) / 2.a
The formula of the solutions is called Quadratic Formula.
NOTE: sqrt(y) means square root of y.
The expression (b^2 - 4.a.c) is called the discriminant of the
equation and determines the nature of the solutions (roots) of the
quadratic equation.
1-. (b^2 - 4.a.c) = 0
In this case the equation has only one root, because results x1 = x2.
The only solution is x = -b/2.a
2-. (b^2 - 4.a.c) < 0
In this case the equation has not a real solution, because there is
not a real number s that satisfies s^2 < 0 (square root is not defined
for negative numbers). The equation's roots are two complex conjugates
numbers.
3-. (b^2 - 4.a.c) > 0
In this case there are two different roots, each one related with the
positive and negative sqrt(b^2 - 4.a.c).
Summing up, the sign of the Discriminant of the equation determines
the nature of the roots of the quadratic equation. If the discriminant
is positive then the equation has two different roots; if the
discriminant is null then the equation has only one root; if the
discriminant is negative the equation has not a solution in the set of
the real numbers, there are two complex conjugates numbers.
For additional reference and more clear equations and formulas, visit
the following pages at SOS Math:
"Quadratic Equations: Quadratic Formula":
http://www.sosmath.com/algebra/quadraticeq/quadraformula/quadraformula.html
"Roots of Quadratic Equations: Summary"
http://www.sosmath.com/algebra/quadraticeq/quadraformula/summary/summary.html
I hope this helps you. If you need further assistance on this, please
let me know by using the clarification feature before rate this
answer.
Best regards.
livioflores-ga |
