
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 x^{2} = 1.
Here we describe: With complex numbers all quadratic equations are solvable. 

Quadratic equations with complex solutions 
We come back to the equation x^{2}= 1. This quadratic equation is not solvable in the domain of real numbers; but in the area of complex numbers.
The equation z^{2}= 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: 

So i is a solution of the equation z^{2} = 1. 
The quadratic equation z^{2} = 1 has two solutions, namely the two conjugate complex numbers z_{1} = i and z_{2} = i. 


As an example, we solve the equation z^{2} = 2. We write: 

The two conjugate complex solutions are: 


The equation is equivalent to the equation: 
with the two solutions and 
and


Completing the square 
The quadratic equations discussed in the previous section, in which z occurred only as z^{2}, were easily solved. In this section we will treat a quadratic equation of the form withreal and . 
As an example we calculate the following 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: 

The equation is equivalent to the equation . The reshaped equation looks no less complicated than before, but it can be solved like the equations discussed in the previous section 
The equation is equivalent to 
If we take the root of it we get or 
The solutions are and 
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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 


The summand 

is taken as the product of a binomial formula and the quadratic addition: 


This results in: 


The equation is thus equivalent to the equation 



Whether the number on the right is positive or negative depends on whether it is positive or negative. The denominator is always positive. The term is 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 and write: 

Then we use a number ω, this means . 
If , then is because .
If , then is and 

For the equation 

we get the two solutions 
and 
ergo and 

We write the short solution: 


Summary 
The solution of the quadratic equation with 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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