Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Introduction to Quadratic Equations and Complex Numbers

The equation  ax^2+bx+c=0 is known as quadratic equation in one variable where  D=b^2-4ac \geq 0 i.e., we get the roots in real numbers.

Let’s take the equation  x^2=-1, where  x=\sqrt{-1}, this is not a real number, i.e., for a quadratic equation  ax^2+bx+c=0 where D=b^2-4ac <  0, we can’t get a real roots for this. So we can take i=\sqrt{-1} , is the imaginary number to find out the root of the equation  x^2=-1.

Hence D=b^2-4ac < 0 , in the quadratic equation  ax^2+bx+c=0 we get two imaginary roots.

Complex Number:

The number which is of the form  a+ib, where  a and  b are real numbers is defined to be a complex number.

We also take z=a+ib , where  a is the real part, denoted by Rez  and  b is the imaginary part denoted by  Imz.

The set consists of all the complex numbers is called a complex number set or set of complex numbers and denoted as  \mathbb{C} .

For example, 

z=5+i7 ,

Here Re~Z=5, ~Im~Z=7

Two complex numbers  z_1=a+ib and  z_2=c+id are equal iff  a=c and  b=d

Example:

If 4x+i (3x-y)=3+i (-6) , where x and  y are real numbers, then find the value of  x and  y.

Solution:

4x+i(3x-y)=3+i(-6) (Given)

\implies 4x=3, ~ 3x-y=-6

\implies x=\dfrac{3}{4} \implies 3\times \dfrac{3}{4}-y=-6 \implies y=\dfrac{33}{4}

\therefore x=\dfrac{3}{4},~~ y=\dfrac{33}{4}

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