Complex Numbers and Quadratic Equations – JEE Mains Mathematics

1. Introduction to Complex Numbers

Complex numbers can be represented as ordered pairs of real numbers (x, y)
Standard form: a + ib, where a is the real part and b is the imaginary part
Visual representation using Argand diagram

2. Algebra of Complex Numbers

Addition, subtraction, multiplication and division of complex numbers
Conjugate of a complex number: If z = a + ib, then z̄ = a - ib
Modulus:
```
|z| = √(a² + b²)
```
Argument (amplitude): Angle made with the positive real axis

3. Quadratic Equations

Definition and standard form:
```
ax² + bx + c = 0
```
Solutions in real and complex number systems
Discriminant:
```
D = b² - 4ac
```

Nature of roots:

```
D > 0: Two distinct real roots
```
```
D = 0: One real root (repeated)
```
```
D < 0: Two complex roots
```

4. Relations and Formations

Relation between roots and coefficients: If α and β are roots:
- ```
α + β = -b/a
```
- ```
αβ = c/a
```
Formation of quadratic equation with given roots:
- If α and β are known, equation is
```
(x - α)(x - β) = 0
```
- Expanding gives:
```
x² - (α + β)x + αβ = 0
```