🔹 1. Introduction

This chapter introduces:

• Complex Numbers (extension of real numbers)
• Quadratic Equations (important algebraic equations)

👉 Very important for:

• Algebra
• Coordinate Geometry
• JEE & CBSE exams

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🔹 2. Complex Numbers

✔️ Definition

A complex number is of the form:

$z = a + ib$z=a+ib

Where:

• a = real part
• b = imaginary part
• i = √−1

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✔️ Important Result

$i^2 = -1$i2=−1

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✔️ Powers of i

• i¹ = i
• i² = −1
• i³ = −i
• i⁴ = 1

(Pattern repeats)

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🔹 3. Algebra of Complex Numbers

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✔️ Addition

$(a + ib) + (c + id) = (a+c) + i(b+d)$(a+ib)+(c+id)=(a+c)+i(b+d)

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✔️ Multiplication

$(a + ib)(c + id) = (ac – bd) + i(ad + bc)$(a+ib)(c+id)=(ac−bd)+i(ad+bc)

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🔹 4. Conjugate of Complex Number

✔️ Definition

$\bar{z} = a – ib$zˉ=a−ib

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✔️ Important Properties

$z \cdot \bar{z} = a^2 + b^2$z⋅zˉ=a2+b2

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🔹 5. Modulus of Complex Number

✔️ Definition

$|z| = \sqrt{a^2 + b^2}$∣z∣=a2+b2​

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✔️ Properties

• |z₁z₂| = |z₁||z₂|
• |z₁/z₂| = |z₁| / |z₂|

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🔹 6. Argand Plane (Concept)

• Real axis → horizontal
• Imaginary axis → vertical
• z = (a, b)

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🔹 7. Quadratic Equations

✔️ General Form

$ax^2 + bx + c = 0$ax2+bx+c=0

$a$a

$b$b

$c$c-10-8-6-4-2246810-10102030-2.002.00

Where a ≠ 0

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🔹 8. Solution of Quadratic Equation

✔️ Formula

$x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}$x=2a−b±b2−4ac​​

$a$a

$b$b

$c$c-10-8-6-4-2246810-10102030-2.002.00

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🔹 9. Discriminant

✔️ Definition

$D = b^2 – 4ac$D=b2−4ac

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✔️ Nature of Roots

• D > 0 → real and distinct
• D = 0 → real and equal
• D < 0 → complex roots

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🔹 10. Relation Between Roots and Coefficients

If roots are α, β:

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✔️ Sum of Roots

$\alpha + \beta = -\frac{b}{a}$α+β=−ab​

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✔️ Product of Roots

$\alpha \beta = \frac{c}{a}$αβ=ac​

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🔹 11. Forming Quadratic Equation

If roots are α, β:

$x^2 – (\alpha + \beta)x + \alpha\beta = 0$x2−(α+β)x+αβ=0

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🔹 12. Complex Roots of Quadratic

If roots are complex:

👉 They occur in conjugate pairs

Example:
a ± ib

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🔹 13. Important Results

✔️ Sum of roots = −b/a
✔️ Product of roots = c/a
✔️ Complex roots come in pairs
✔️ Discriminant determines nature

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🔹 14. Applications

✔️ Algebraic equations
✔️ Physics problems
✔️ Engineering calculations
✔️ Graph analysis

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🔹 15. JEE & CBSE Important Points

✔️ Discriminant-based questions are common
✔️ Practice root relations
✔️ Formation of equations important
✔️ Complex numbers basics frequently asked
✔️ Learn properties of modulus

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🔹 16. Common Mistakes

❌ Sign errors in formula
❌ Wrong calculation of D
❌ Ignoring conjugate roots
❌ Mistakes in simplification

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🔹 17. Practice Questions

1. Solve quadratic equation
2. Find nature of roots
3. Form equation from roots
4. Simplify complex numbers
5. Find modulus and conjugate

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🔹 18. Quick Revision Sheet

• z = a + ib
• |z| = √(a² + b²)
• i² = −1
• ax² + bx + c = 0
• D = b² − 4ac
• α + β = −b/a
• αβ = c/a

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🔹 19. Conclusion

Complex Numbers & Quadratic Equations is a fundamental and high-weightage chapter.

👉 Strong concepts here help in:

• Higher algebra
• Calculus
• Competitive exams like JEE