COMPLEX NUMBERS IN QUADRATIC EQUATIONS

A quadratic equation is an equation where the maximum degree of the equation is two. 

The standard form of a quadratic equation is ax2+bx+c=0, where x is a variable and a, b and c are real numbers and a≠0. 

The roots of the given equation are   

x=-bb2-4ac2a

Here, Discriminant(D)= b2-4ac

When D=0, then the roots of the quadratic equation are real and equal. 

When D >0, then roots of the quadratic equation are real and unequal.

When D<0, then the roots of a quadratic equation are non-real(complex).

The number in the form of a±ib, where a and b are real numbers are called complex numbers. Here a is the real part and b is the imaginary part of the complex number.

For example,equation x2+1=0 has no real solution. x2=-1 and the square of every real number is non-negative. Therefore x=±1 is the solution of this equation.

COMPLEX NUMBERS AND QUADRATIC EQUATIONS

Let -1=i

        i2=-1  

means i is the solution of the equation x2+1=0

For complex number z=a+ib, where a is the real part denoted by Re z and b is the imaginary part 

 

ALGEBRA OF COMPLEX NUMBERS-

The algebra of complex numbers are-

  1. Addition of two Complex Numbers-

Let z1=a+ib and z2=c+id be any two complex numbers. Then sum z1+ z2 is given by 

 z1+z2=a+c+i (b+d), which is again a complex number.

  1. Difference of two complex Numbers-

The difference of z1-z2 is given by 

   z1-z2=z1+-z2

Let z1=a+ib and z2=c+id be any two complex numbers. Then difference z1- z2 is given by 

 z1-z1=a-c+i (b-d), which is again a complex number.

  1. Multiplication of two complex numbers-

Let z1=a+ib and z2=c+id be any two complex numbers. Then the product z1*z2 is given by           z1*z2=ac-bd+i(ad+bc)

  1.  Division of Complex number-

Let z1=a+ib and z2=c+id be any two complex numbers, where z2≠0, then z1z2 is given by

  z1z2=a+ibc+id

  1. Square root of a negative real number 

The square root of –1 are i and –i. 

    MODULUS AND CONJUGATE OF THE COMPLEX NUMBER-

     Let z=a+ib be the complex number. Then, the Modulus of z is denoted by 

      |z|, is defined to be the non-negative real number a2+b2

      i.e.    z=√(a2+b2) and conjugate of z, denoted by z̅ is the complex number a-ib

       i.e.     z=a-ib

     For any two complex numbers, z1 and z2 , we have

      z1z2=z1|z2|

      z1z2=z1z2  provided |z2|≠0

 POLAR REPRESENTATION OF COMPLEX NUMBERS-

Let P be the point representing a non-zero complex number z=x+iy. Point P can be determined by ordered pair of real numbers (r, ϴ) called polar coordinates of point P.  

 x=r cosϴ and y=r sinϴ, therefore z=r(cosϴ+isin ϴ)

Hence, this is called a polar form of a complex number.

CONCLUSION –

The complex numbers are  given by z=a+ib, where a is the real part and b is the imaginary part. When the discriminant of quadratic equation is negative, then the roots are non-real or complex. The solution for complex root in can be represented in polar form by z=rcosϴ±i sinϴ. Above are listed rules to calculate complex numbers. 

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