Evaluation of Complex Numbers and Quadratic Equations

Complex Numbers and Quadratic Equations

How can the x2+1=0 type equation be solved? If we solve this equation, we get x2=-1, but the square of every real number is non-negative. So, we need to increase the real number system to a larger system to find the solution of the equation, i.e., x2+1= 0, for the real number. Now, we will study the system of Complex numbers to resolve this equation problem.

Complex Numbers 

In this section, we will read about complex numbers. √-1 by the symbol of I, or we call it iota.

Then, we have i2=-1, which means i is the solution of the equation x2+1 = 0.

A complex number is in the form of a+ib where a is the real number and b is the imaginary number, and we can say that a is the real part of z (complex number) and b is the complex part of z.

For example 5 +3i where Re z = 5 and Im z = 3

Algebra of Complex Numbers

Let z1 = a + ib and z2 = c + id, then the sum of two complex numbers (z1+ z2) can be calculated as:    

 z1+ z2 = (a + ib) + (c + id)

=(a + c) + i(b + d)

Therefore, 

z1 + z2 = Re (z1+ z2) + Im(z1+ z2) 

The Addition of Complex Numbers satisfy the following properties:

Closure Law: The sum of two complex numbers is also a complex number. For z1 + z2, where z1 and z2 are complex numbers, the total sum z will also be a complex number.

Commutative Law: As per commutative law, for any two complex numbers like z1 and z2:

z1 + z2 = z2 + z1.

Associative Law: For any three complex numbers (z1+ z2 )+ z3 = z1+ (z2+ z3).

Existence of Additive Identity: Additive identity is denoted as 0 , such that, for every complex number z, z + 0 = z.

Existence of Additive Inverse: Additive inverse, or negative of any complex number z, is a complex number whose real and imaginary parts have the opposite sign. It is represented by –z and z + (-z) = 0

Subtraction of two Complex Numbers

Let z1= a + ib and z2 = c + id, then the difference of the two complex numbers (z1 – z2) is:

z1- z2= (a + ib) – (c + id)

= (a – c) + i (b – d)

So, z1 – z2 = Re(z1 – z2) + Im(z1-z2)

Multiplication of two Complex Numbers

Let z1= a + ib and z2 = c + id, then the multiplication of the two complex numbers (z1× z2) is:

z1× z2 = (a + ib) ×(c + id). where i2= -1

z1×z2= (ac – bd) + i(ad + bc)

Closure, Commutative and Associative laws are the same for multiplication as in the case of the addition of complex numbers.

Multiplicative Identity: Multiplicative Identity is denoted as 1 (1 + i0), for every complex number z, z.1 = z.

Multiplicative Inverse: For any non-zero complex number z, 1/z, or z-1 is multiplicative inverse as z×1/z = 1 

Distributive Law: For any three complex numbers z1, z2, and z3 –

z1 (z2+ z3 )= z1 z2+ z1 z3

Division of two Complex Numbers

Let z1 = a + ib and z2 = c + id, then the division of two complex numbers (z1/z2) is also a complex number.

It is denoted as: a+ib /c+id

Modulus and Conjugate of Complex Number

If the complex number z = a+ib, then |z| is called the modulus of a complex number.

|z|= a2+b2

For example |6-7i|=62+(-7)2=36+49=85

Conjugate of Complex Number

We can find the complex conjugate by changing the sign of the imaginary part of the complex number. 

Example:

The complex conjugate of 4+7i is 4-7i. 

Quadratic Equation 

Quadratic equation is of the type of ax2+bx+c=0, with real coefficient a,b,c, and a ≠ 0

When b2-4ac<0 then the solution of the equation is

x= -b±√(b2-4ac)/2a

For example: x2 +3x+5=0

Where a=1, b=3, c=5

b2-4ac=9-20=-11 which is less than 0

So, the solution of equation is x=-3 ± √-11/2

Conclusion

From the information mentioned above, we can find the solution of the equation of the type x2+1=0. A complex number is written in the form of a+ib, where a is the real part and b is the imaginary part of the complex number, z. Here, we discussed the algebra of complex numbers. If we want to find the solution to the Quadratic equation, then we can use x= -b±(√b2-4ac)/2a.