Quadratic Equations in a complex number system

A polynomial expression is an expression of the form of  a0x0 + a1x1+ ……. anxn (n ∊ N), and an is not zero. Here, n is the degree of this polynomial. When n is 2, this expression is quadratic. An equation of the form a0x0 + a1x1 + a2x2 = 0 is a quadratic equation. It will always have 2 roots/zeros/solutions. Learn more about the basics of quadratic equations with complex numbers and easy tips and tricks to solve Quadratic equations examples.

Introduction

Depending on the coefficients, a quadratic equation may have none, one or two real roots. Some roots might consist of imaginary numbers and these are referred to as complex roots. That means, the solution of such equations includes complex numbers. In this article, we will cover the generality of quadratic equations, quadratic equations with complex numbers in detail. At the same time, we will discuss quadratic equations and complex numbers individually before explaining how to solve quadratic equations in a complex number system. 

What is a Quadratic Equation?

A quadratic equation is a mathematical expression of degree 2. Fundamental Theorem of Algebra states that

 “A polynomial equation of degree n has exactly n roots”

It must be noted that the roots can be real, complex as well as there can be repeated roots. Any equation that is written in a standard form of a1x(n-1) + a2x(n-2) + a3x(n-3) + … + an = 0 is a polynomial equation of degree ‘n’, thus, it has ‘n’ roots. Thus, a quadratic equation that is a polynomial equation of order 2 has two solutions.

Another, more used form of the quadratic equation is ax² + bx + c = 0.

a,b and c are the constant coefficients, x is the variable. The first condition for an equation to be a quadratic equation is that the coefficient of x2 is non-zero i.e. a≠0. 

Not only a but b and c can also be complex, even if they are real, the equation may have complex roots, in which case the roots are conjugate pairs.

Quadratic equations in a complex number system

For a quadratic equation 𝑎𝑥2+𝑏𝑥+𝑐=0 with 𝑎≠0, the solutions/roots are given by 𝑥=(−𝑏±√(𝑏2−4𝑎𝑐))/a. This is known as the quadratic formula.

The term b2 – 4ac is referred to as the discriminant of the equation. Depending on its sign, the equation may have real, complex, or coincident roots.

Using the discriminant, we identify the three different cases of quadratic equations as follows( note that we are considering only real a, b and c):

1. Positive discriminant: 𝑏2−4𝑎𝑐>0, two real roots

2. Zero discriminant: 𝑏2−4𝑎𝑐=0, one repeated real root

3. Negative discriminant: 𝑏2−4𝑎𝑐<0, no real roots or, as we say, two distinct complex roots

Thus it is quite obvious that the total roots are two in each case.

Hence to know Quadratic equations in a complex number system – the discriminant must be NEGATIVE.

For the equation ax2+bx+c=0

If α,β are the roots of the quadratic equation, then the sum of the roots (α+β)= -b/a and the product of the roots (α.β)= c/a

Additionally, P and S are the product of roots and sum of roots, respectively, of a quadratic equation. The quadratic equation is then given by x2– Sx + P = 0.

 What are Complex Numbers?

A complex number can be represented in the form of a+ib. Here, a and b are real numbers, and i is the imaginary constant iota, such that i2= -1. Hence, a complex number is the sum of a real and imaginary number. Additionally, complex numbers are used to find the square root of negative numbers.

The concept of complex numbers came into the picture because of a few doubts, such as; what if the solution of an equation does not lie on the number line? For example, what is the square root of a negative number? And so on.

Naturally, the square of a real number is always non-negative, e.g. (4)2 = 16 and (– 4)2 = 16. Therefore, the square root of 16 is ± 4. 

But for the square root of a negative number, Euler (1707 – 1783) introduced the symbol i (iota) for the positive square root of – 1, i.e., i = −1.

I = −1, i2 = –1, i3 = i2i = –i, i4 = (i2)2 = (–1)2 = 1.

This allowed us to find and analyze the roots of the equations which had no real solution.

It should be understood that for most identities, an imaginary number acts the same as a real number.

Let us understand the concept with the help of examples.

Question – Express (5-3i)² in the form of a+ib

Solution

(5-3i)² = 52 -2*5*3*i + (3i)2

= 16 – 30i

Examples of Quadratic equations of complex numbers

Question – Find all the roots, real and complex, of the equation x3 – 2x2 + 25x – 50 = 0.

Solution

First, factor the equation to get x2(x – 2) + 25(x – 2) = (x – 2)(x2 + 25) = 0. 

Using the multiplication property of zero, you determine that x – 2 = 0 and x = 2. 

You also get x2 + 25 = 0 and x2 = –25. 

Take the square root of each side, and

Simplify the radical, using the equivalence for i, and the complex solutions are

The real root is 2, and the imaginary roots are 5i and –5i.

Question

Find the roots: x2 + 4x + 5 = 0

Solution

Being non-factorable, we apply the quadratic formula. After combining the values, there is a negative value under the square root radical. This negative square root creates an imaginary number.

The complex roots in this example are x = -2 + i and x = -2 – i. These roots are similar except for the “sign” separating the two terms. One root is -2 Plus I, and the other is -2 Minus i. When solving a quadratic equation, such a pattern will be discovered in every set of complex roots. Roots that possess this pattern are called complex conjugates (or conjugate pairs).

CONCLUSIONS

We saw the basic structure and definition of a polynomial equation, a special case of which is a quadratic equation. We also learned the claims of the fundamental  theorem of arithmetic, which claims that a quadratic equation must have two roots. On this basis, we saw the different possible scenarios of the nature of roots. We also learned the quadratic formula, which is a direct method to find the roots of a quadratic equation.