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-
- 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.
- 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.
- 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)
- 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
- 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.