Solving Quadratic Equations with Complex Roots
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When the roots of a quadratic equation are imaginary,
they always occur in conjugate pairs.

 A root of an equation is a solution of that equation. 


If a quadratic equation with real-number coefficients
has a negative discriminant,
then the two solutions to the equation are complex conjugates of each other.
(Remember that a negative number under a radical sign yields a complex number.)

The discriminant is the  b2- 4ac  part of the quadratic formula (the part under the radical sign). 
If the discriminant is negative, when you solve your quadratic equation the number under the radical sign in the quadratic formula is negative --- forming complex roots.

                             Quadratic equation:  
 

                              Quadratic formula:   
    

 

Example 1:

Find the solution set of the given equation over the set of complex numbers.

a = 1,     b = -10,     c = 34

Pick out the coefficient values representing a, b, and c, and substitute into the quadratic formula, as you would do in the solution to any normal quadratic equation.

Remember, when there is no number visible in front of the variable, the number 1 is there.


 

HINT:  When the directions say:
 Express over the set of complex numbers,
look for a negative value under the radical sign.

 

     

Example 2:

               Find the solution set of the given equation over the set of complex numbers.

a = 3,    b  = -4,   c  = 10

 

    

Example 3:

Find the solution set of the given equation and express its roots in a+bi form.

 

 


*
Be sure to set the quadratic equation equal to 0.

* Arrange the terms of the equation from the
   highest exponent to the lowest exponent.