# Complex Numbers and Quadratic Equations Notes | SwaVid

Complex Numbers and Quadratic Equations, NCERT Class 11 Mathematics: notes with key ideas, formulas, common mistakes and 3 quick checks with worked answers.

Canonical: https://www.swavid.com/cbse/class-11/maths/chapter/complex-numbers-and-quadratic-equations

Source: https://www.swavid.com/cbse/class-11/maths/chapter/complex-numbers-and-quadratic-equations

# Complex Numbers and Quadratic Equations

## Formulas and definitions

## Key ideas

## Common mistakes

## Quick checks

## Keep going

## Questions parents ask

## Class 11 Mathematics on SwaVid

### Need for complex numbers

### Definition of a complex number

### Algebra of complex numbers

### Modulus and conjugate

### Multiplicative inverse

### Quadratic equations with negative discriminant

### Next chapter

### Related chapter

### All chapters

SwaVid › CBSE › Class 11 › Mathematics › Complex Numbers and Quadratic Equations

Notes written by SwaVid from the NCERT textbook.

Class 11 Mathematics, Chapter 4. Key ideas, formulas and common mistakes.

CBSE does not print marks for a single chapter.

Short answer. This chapter introduces complex numbers to solve quadratic equations that have no real roots. Students learn algebraic operations on complex numbers, modulus, conjugate, and finding complex solutions of quadratic equations.

Source: NCERT Mathematics (Reprint 2026-27) , Chapter 4, Complex Numbers and Quadratic Equations

The equation x² + 1 = 0 has no solution in real numbers because the square of every real number is non-negative. To overcome this, the imaginary unit i is introduced such that i² = -1.

A number of the form a + ib, where a and b are real numbers, is called a complex number. Here, a is the real part denoted as Re(z), and b is the imaginary part denoted as Im(z).

Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. Addition, subtraction, multiplication, and division follow regular algebraic rules, using i² = -1.

For a complex number z = a + ib, the modulus |z| is √(a² + b²) and the conjugate z̄ is a - ib. They satisfy the fundamental relation z × z̄ = |z|².

The multiplicative inverse of a non-zero complex number z = a + ib is z⁻¹ = z̄ / |z|². It equals (a - ib) / (a² + b²).

For ax² + bx + c = 0 with real coefficients where b² - 4ac < 0, the roots are complex numbers. The solutions are given by x = (-b ± i√(4ac - b²)) / (2a).

Try each question first. Then tap it to see the answer.

(2 - 3i)(1 + 2i) = 2(1 + 2i) - 3i(1 + 2i)

= 2 + 4i - 3i - 6i²

= 2 + i - 6(-1)

= 2 + i + 6

= 8 + i

8 + i

|z|² = 4² + (-3)² = 16 + 9 = 25

z̄ = 4 + 3i

z⁻¹ = z̄ / |z|² = (4 + 3i) / 25

= 4/25 + (3/25)i

4/25 + (3/25)i

Here a = 1, b = 2, c = 5

Discriminant D = b² - 4ac = 2² - 4(1)(5) = 4 - 20 = -16

√D = √(-16) = 4i

x = (-2 ± 4i) / (2 × 1)

x = -1 ± 2i

-1 ± 2i

Linear Inequalities .

Trigonometric Functions .

The 14 chapters of Class 11 Mathematics .

The CBSE Class 11 Maths curriculum says it gives no chapter or unit-wise weightage. Theory is 80 marks, plus 20 for internal assessment.

No. They are written by SwaVid from the NCERT textbook. The CBSE curriculum is the reference for what is assessed.

Linear Inequalities.

SwaVid is an AI tutor for Indian students in Classes 6 to 12. It finds the earlier ideas a child missed, then teaches today’s chapter. Start free to see how it works.

- Ch 4 NCERT book
- 80 marks, theory paper
- 3 quick checks
- Writing the identity √(a) × √(b) = √(ab) when both a and b are negative . The identity √(a) × √(b) = √(ab) holds only if at least one of a or b is non-negative. For √(-1) × √(-1), write i × i = -1, not √((-1)(-1)) = 1.
- Including the symbol i when writing the imaginary part . The imaginary part of a + ib is the real number b, not ib. For example, for z = 3 + 4i, Im(z) = 4.
- Adding a negative sign inside the square in the modulus formula . The modulus |a + ib| is √(a² + b²). For 3 - 4i, |z| = √(3² + (-4)²) = √(9 + 16) = 5, not √(9 - 16).

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