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Quadratic Formula

A quadratic equation with real or complex coefficients has two (not necessarily distinct) solutions, called roots, which may or may not be real, given by the quadratic formula:
 
where the symbol "±" indicates that both
 are solutions.
Simply put, ± means 'plus or minus' as equation possibilities.
 
Discriminant:  is called the discriminant of the quadratic equation.
 
A quadratic equation with real coefficients can have either one or two distinct real roots, or two distinct complex roots. In this case the discriminant determines the number and nature of the roots. 
 
There are three cases:
 
Real Roots: If the discriminant is positive, there are two distinct roots, both of which are real numbers. For quadratic equations with integer coefficients, if the discriminant is a perfect square, then the roots are rational numbers-in other cases they may be quadratic irrationals.
 
Equal Roots: If the discriminant is zero, there is exactly one distinct root, and that root is a real number. Sometimes called a double root, its value is: 
 
Imaginary Roots: If the discriminant is negative, there are no real roots. Rather, there are two distinct (non-real)complex roots, which are complex conjugates of each other:
 


 
Thus the roots are distinct if and only if the discriminant is non-zero, and the roots are real if and only if the discriminant is non-negative. 
  • Sum of the roots of a quadratic equation = -b/a
  • Product of the root of a quadratic equation = c/a
If a and b are roots of equation then D = 
 
Now there are three conditions
  • If D < 0, then roots are imaginary.
  • If D > 0, then roots are real
  • If D = 0 then roots are equal




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