Complex Numbers and Quadratic Equations

Definition of Complex Numbers

Before Defining complex numbers, assume that  √-1 = i or i² = -1 which means i can be assumed as the solution of the equation x² + 1 = 0. i is called as Iota in complex numbers.

We can further formulate as,

i² = -1

i³ = i² * i = -i

i⁴ = i² * i² = 1

So, we can say now, i⁴ⁿ = 1, where n is any positive interger.

Also, note that i + i² + i³ + i⁴  = 0 or iⁿ + i²ⁿ + i³ⁿ + i⁴ⁿ = 0

Now, any number of the form a + ib, where a and b are real numbers are defined as complex numbers. For example: 4 + 5i, -3 +23i, -3 + i√5 are complex numbers.  

Complex numbers are denoted by z or w (= a + ib) which has two parts, one is called as the real part and another as imaginary part. z= a (or b = 0) is called as the purely real number while z = ib ( or a = 0) is called as the purely imaginary number. Also note that zero or 0 + i0 is both purely real and purely imaginary but not imaginary.

Real part of z is a, denoted by Re(z) and Imaginary part of Z is b, denoted by Imag(z). For z = 1+2i, re(z) = 1 and Imag (z) = 2

Complex number is considered as the super-set of all the other possible numbers.

N⊂ W ⊂ I⊂Q ⊂R ⊂C 

Definition of Quadratic Equation

An Algebraic Equation with degree n is called as the Polynomial Equation. Quadratic equation is the special case of it where the degree is equals to 2. So we can say that a polynomial with degree 2 or two roots (or solution) is called as the quadratic equation.

The general form of any Quadratic equation is as y = ax²+ bx + c, where a ≠ 0, b and c can be any real (or complex) number. Here, a, b and c are called as the Coefficients. a is also called as the Leading Coefficient.

Shape of the quadratic equation is parabolic in nature. For y = ax²+ bx + c, the parabola is open upwards if a is positive and open downwards if a is negative. While for x = ay²+ by + c, parabola is open rightwards if a is positive while open open leftwards if a is negative.

Quadratic equation has two roots or solution because of its degree 2. These two roots are given with the help of Quadratic Formula as:

Roots of any equation indicates the point on the x-axis where the the curve for any function cuts the x - axis. The number of points where the curve cut the x-axis denoted the number of roots or solution or degree of the given polynomial.

The graph for the quadratic equation is in Parabolic Shape. If a is positive, i.e. a > 0, then the parabola will be open-upwards and hence will only have a least value or minimum value of this function while for a to be negative, i.e. a < 0, the parabola will be open-downwards and will have a definite maximum value but not minimum value.
 

What is Complex Root? Or what is Complex solution?

Roots for any equation are the set of points where the graph for the given function intersects or touches the x–axis. The number of roots can be directly related with degree of a polynomial.

But if the graph of a polynomial does not intersect, in such cases we assume as the complex root (or solution). This means that, there no such real number but the complex number which will satisfy the given functional equation.

While solving the Quadratic equation, if b² - 4ac < 0, then the roots will not be real roots and the roots in this case will be called as the Complex Roots.
 

What are the different forms of representation for complex numbers?

Complex numbers can be represented into mainly four forms. They are:

• Cartesian or algebraic or rectangular form

• Trigonometric or polar form

• Exponential form

• Vector form
   

What is the discriminant of a quadratic equation?

The value of D = b² – 4ac, in quadratic means discriminant. Discriminant plays most important role in deciding the nature of the roots. It is also represented by delta or ∆.

If D = 0, means equal roots. Graphically, parabola touches the x-axis at a single point.

If D > 0, means real and distinct roots, Graphically, parabola intersects the x-axis at two distinct points.

And if D < 0, means non-real or imaginary or complex roots. In this case the parabola never touches the x-axis.

What are the different ways of solving quadratic equation?

Solving Quadratic equation means finding the values of the variable which satisfies the given equation. Quadratic equation can be solved in two ways. The first method is called the Factorization Method. While the second method is called as the Hindu Method or Sri dharachary method. This method (Hindu) is formula based method.
 

Few Examples of Complex Numbers and Quadratic Equations

Complex Numbers:

Complex Numbers has wide range of applications but most importantly used in the field of electrical engineering. In Electrical engineering, there are many applications which the complex numbers comes into the picture. Such as, design of circuits using Capacitor and inductors, electromagnetism, oscillations and many more.

Quadratic Equations:

A quadratic equation traces the parabolic cubic curve. We come across many real situations in which the path or the equation follows the parabolic curve i.e. they can be easily related with the help of quadratic equation. A boll thrown in air or jumping of any person from the building are some of the examples where the path traced by them in their journey follows the parabolic curve. Understanding quadratic equation helps us in understanding their instantaneous location during their journey.

It also helps in figuring out the profit of any business using quadratic equations.

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Complex Numbers and Quadratic Equations