RedCrab Math Tutorial
 



Quadratic Equations & Complex Numbers

This article describes an important application of complex numbers, the solution of quadratic equations. An example of the solvability of a quadratic equation has already been described in the article Complex Numbers, namely x2 = -1.

Here we describe: With complex numbers all quadratic equations are solvable.

 

Quadratic equations with complex solutions

We come back to the equation x2= -1. This quadratic equation is not solvable in the domain of real numbers; but in the area of complex numbers.

The equation z2= -1 has the complex solution z = i. The unknown z instead of x indicates that a complex solution is allowed.

 
Obviously, the following solution is also correct:
quadratic Equations Solution
So -i is a solution of the equation z2 = -1.
The quadratic equation z2 = -1 has two solutions, namely the two conjugate complex numbers z1 = i and z2 = -i.
 
 
As an example, we solve the equation z2 = -2. We write:
konjugiert komplexe Zahl
The two conjugate complex solutions are:
konjugierte komplexe Lösung
 
The equation quadratische Gleichung is equivalent to the equation:
äquivalente komplexe Lösungwith the two solutions erste komplexe Lösungand zweite komplexe Lösung

komplexe Lösung 1 andkomplexe Lösung 2

 

Completing the square

The quadratic equations discussed in the previous section, in which z occurred only as z2, were easily solved. In this section we will treat a quadratic equation of the form quadratisch ergänzenwithreal and .

As an example we calculate the following quadratic equation:

quadratic equation

We try to apply the solution method from the previous section to this equation.

To write the left side of the equation as a square, we use the method of quadratic addition. In our example, this looks like this:

Completing the square

The equation  is equivalent to the equation umgeformte Gleichung. The reshaped equation looks no less complicated than before, but it can be solved like the equations discussed in the previous section
The equation(z-3)^2+4=0 is equivalent to(z-3)^2=-4=4*(-1)
If we take the root of it we get z-3=2ior z-3=-2i
The solutions are z1=3+2i and z2=3-2i
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General solution formula for quadratic equations

We again use the general quadratic equation of the form
 
A division by a results
General solution formula
 
The summand
Summand
is taken as the product of a binomial formula and the quadratic addition:
Quadratic addition
 
This results in:
Quadratic equation
 

The equation quadratisch ergänzen is thus equivalent to the equation

equivalent equation

 

Whether the number on the right is positive or negative depends on whether it is b2-4acpositive or negative. The denominator (2a)^2 is always positive. The term Diskriminanteis the discriminant of the quadratic equation. It therefore depends on the sign of the discriminant whether the quadratic equation has a real or not a real solution.

To solve the equation, we set D for b^2-4ac and write:

Diskriminante Formel

Then we use a number ω, this means .

If , then is because .

If D=-2, then is and

 
For the equation
Diskriminante der quadratischen Gleichung
we get the two solutions
and
ergo and
 
We write the short solution:
 

Summary

The solution of the quadratic equation quadratisch ergänzenwith real and is:

The complex number ω is:
Depending on the sign of the discriminant D, the solutions are as follows:

If D > 0, then are z1 and z2 real

If D = 0, then are z1 = z2 real

If D < 0, then are z1 and z2 conjugated complex to each other

 
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Complex numbers

Basics of complex numbers
Computing with complex numbers
Addition and subtraction
Multiplication of complex numbers
Conjugate complex numbers
Division of complex numbers
Quadratic equations, complex numbers
Geometric representation
Geometric addition and subtraction
Absolute value of a complex number
The polar form of complex numbers
Calculate polar form
Convert polar form to normal
Multiplication in polar form
Division in polar form
   


           
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