Quadratic Equations

“A quadratic equation is an equation of the form ax² + bx + c = 0, where a ≠ 0.”

Key Concepts

  • Standard Form: ax² + bx + c = 0
  • Roots of Quadratic Equation: The values of x that satisfy the equation.
  • Discriminant (D): D = b² - 4ac helps determine the nature of roots.
  • Nature of Roots:
    • D > 0: Real and distinct roots
    • D = 0: Real and equal roots
    • D < 0: Imaginary roots

Formulas

  • Quadratic Formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
  • Sum of Roots: -b/a
  • Product of Roots: c/a

Solved Examples

  • Example 1: Solve: x² - 5x + 6 = 0
  • Solution: Factorizing, (x - 2)(x - 3) = 0 ⇒ x = 2, 3
  • Example 2: Solve using formula: 2x² - 4x + 1 = 0
  • D = (-4)² - 4×2×1 = 16 - 8 = 8 ⇒ x = [4 ± √8]/4 = [4 ± 2√2]/4 = 1 ± √2/2

Previous Year Questions

  • 1. SSC CGL 2020: Find the roots of x² + 7x + 10 = 0
  • Roots: (x + 2)(x + 5) = 0 ⇒ x = -2, -5
  • 2. SSC CHSL 2021: What is the sum and product of roots of x² - 4x + 3 = 0?
  • Sum = 4, Product = 3
  • 3. RRB JE 2019: Solve: x² + 4x + 5 = 0
  • D = 16 - 20 = -4 ⇒ Roots are imaginary: x = -2 ± i
  • 4. SSC MTS 2018: Factorize: x² - 9x + 20
  • x = 4, 5
  • 5. HSSC 2021: If the roots are equal, find the value of k: x² + 2x + k = 0
  • D = 4 - 4k = 0 ⇒ k = 1
  • 6. SSC CGL 2019: Solve: 3x² - 5x + 2 = 0
  • x = 1, 2/3
  • 7. SSC GD 2021: Find the value of m if roots are equal: mx² + 6x + 1 = 0
  • D = 36 - 4m = 0 ⇒ m = 9
  • 8. SSC CPO 2020: Roots of x² + 10x + 25 = 0?
  • x = -5, -5
  • 9. SSC CHSL 2017: What is the nature of the roots of x² - x + 1 = 0?
  • D = 1 - 4 = -3 ⇒ Imaginary roots
  • 10. SSC CGL 2022: If x² - 8x + 16 = 0, find the value of x
  • x = 4, 4
  • 11. RRB NTPC 2019: Factorize x² - 3x - 10
  • x = 5, -2
  • 12. SSC JE 2018: Find product of roots of 2x² + 5x + 3
  • Product = 3/2
  • 13. SSC CHSL 2022: Solve x² + 3x + 2 = 0
  • x = -1, -2
  • 14. SSC GD 2023: Find sum of roots of 4x² - x - 5
  • Sum = 1/4
  • 15. SSC CGL 2017: Solve: x² + x - 6 = 0
  • x = 2, -3
  • 16. SSC MTS 2019: Discriminant of x² + 2x + 4?
  • D = -12 ⇒ Imaginary roots
  • 17. SSC CGL 2020: Value of k if roots of x² - kx + 16 are equal
  • D = k² - 64 = 0 ⇒ k = ±8
  • 18. RRB ALP 2018: Solve: x² - 3x + 2 = 0
  • x = 1, 2
  • 19. SSC CGL 2023: Roots of 6x² + 11x - 35 = 0
  • x = 1.67, -3.5
  • 20. HSSC CET 2022: If x² + x - 2 = 0, find x
  • x = 1, -2
Note: Always check the discriminant before solving to know the nature of the roots. Use factorization for simple equations and quadratic formula for complex ones.

Conclusion

Understanding quadratic equations is essential for competitive exams. Mastery of factorization, discriminants, and the quadratic formula helps solve these problems quickly and accurately.

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