IIT JEE Main Maths -Unit 2- Quadratic equations in real and complex systems- Study Notes-New Syllabus
IIT JEE Main Maths -Unit 2- Quadratic equations in real and complex systems – Study Notes – New syllabus
IIT JEE Main Maths -Unit 2- Quadratic equations in real and complex systems – Study Notes -IIT JEE Main Maths – per latest Syllabus.
Key Concepts:
- Quadratic Equations in Real and Complex Systems
Quadratic Equations in Real and Complex Systems
A quadratic equation is an equation of the form: 
\( ax^2 + bx + c = 0 \), where \( a, b, c \in \mathbb{R} \) (or \( \mathbb{C} \)), and \( a \ne 0 \).
The solutions (or roots) of the equation are the values of \( x \) that satisfy it.
1. Roots of a Quadratic Equation
The roots are given by the quadratic formula:
\( x = \dfrac{-b \pm \sqrt{b^2 – 4ac}}{2a} \)
Here, the expression \( D = b^2 – 4ac \) is called the discriminant.
2. Nature of Roots (Real System)
| Discriminant \(D\) | Nature of Roots | Remarks |
|---|---|---|
| \( D > 0 \) | Real and distinct | Two different real roots |
| \( D = 0 \) | Real and equal | Both roots are the same |
| \( D < 0 \) | Complex conjugate | Non-real roots |
3. Roots in Complex System
When \( D < 0 \), roots are complex. Let \( D = -k \) where \( k > 0 \).
\( x = \dfrac{-b \pm i\sqrt{k}}{2a} \)
The roots are conjugate pairs of the form \( p + iq \) and \( p – iq \).
4. Relationship Between Coefficients and Roots
If \( \alpha \) and \( \beta \) are the roots of \( ax^2 + bx + c = 0 \), then:
- Sum of roots: \( \alpha + \beta = -\dfrac{b}{a} \)
- Product of roots: \( \alpha \beta = \dfrac{c}{a} \)
Example
Find the roots of \( 2x^2 – 3x + 1 = 0 \).
▶️ Answer / Explanation
Step 1: Identify coefficients \( a = 2, b = -3, c = 1 \).
Step 2: \( D = (-3)^2 – 4(2)(1) = 9 – 8 = 1 \)
Step 3: \( x = \dfrac{3 \pm \sqrt{1}}{4} \)
\( \Rightarrow x = 1 \) or \( x = \dfrac{1}{2} \)
Answer: Roots are \( 1 \) and \( \dfrac{1}{2} \).
Example
Find the roots of \( x^2 + 4x + 8 = 0 \).
▶️ Answer / Explanation
Step 1: \( a = 1, b = 4, c = 8 \)
Step 2: \( D = 4^2 – 4(1)(8) = 16 – 32 = -16 \)
Step 3: \( x = \dfrac{-4 \pm i\sqrt{16}}{2} = \dfrac{-4 \pm 4i}{2} \)
\( \Rightarrow x = -2 + 2i, \; x = -2 – 2i \)
Answer: Complex conjugate roots are \( -2 \pm 2i \).
Example
Find the roots of \( 3x^2 + 2x + 5 = 0 \) and express in the form \( p \pm iq \).
▶️ Answer / Explanation
Step 1: \( a = 3, b = 2, c = 5 \)
Step 2: \( D = b^2 – 4ac = 4 – 60 = -56 \)
Step 3: \( x = \dfrac{-2 \pm i\sqrt{56}}{6} = \dfrac{-2 \pm i2\sqrt{14}}{6} \)
\( \Rightarrow x = \dfrac{-1}{3} \pm i\dfrac{\sqrt{14}}{3} \)
Answer: \( x_1 = -\dfrac{1}{3} + i\dfrac{\sqrt{14}}{3} \), \( x_2 = -\dfrac{1}{3} – i\dfrac{\sqrt{14}}{3} \).
Notes and Study Materials
- Concepts of Quadratic Equations
- Quadratic Equations Master File
- Quadratic Equations Revision Notes
- Quadratic Equations Formulae
- Quadratic Equations Reference Book
- Quadratic Equations Past Many Years Questions and Answer
Examples and Exercise
- Practice Paper 1
- Practice Paper 2
- Practice Paper 3
- Practice Paper 4
- Practice Paper 5
- Practice Paper 6
- Practice Paper 7
- Practice Paper 8
IIT JEE (Main) Mathematics ,”Quadratic Equations” Notes ,Test Papers, Sample Papers, Past Years Papers , NCERT , S. L. Loney and Hall & Knight Solutions and Help from Ex- IITian
About this unit
Quadratic equations in real and complex number system and their solutions. The relation between roots and coefficients, nature of roots, the formation of quadratic equations with given roots.
IITian Academy Notes for IIT JEE (Main) Mathematics – Quadratic Equations
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S. L. Loney IIT JEE (Main) Mathematics
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Hall & Knight IIT JEE (Main) Mathematics
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