Summary of Quadratic Equations

• A quadratic equation is of the form ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0.
• A real number α is a root of the equation if aα² + bα + c = 0.
• The roots can be found by factorizing the equation into linear factors.
• Quadratic Formula: The roots are given by x = (-b ± √(b² - 4ac)) / (2a).
• Nature of roots based on the discriminant D = b² - 4ac:
  • Two distinct real roots if D > 0.
  • Two equal roots if D = 0.
  • No real roots if D < 0.

Quadratic Equations

Introduction

• A quadratic equation in the variable x is of the form ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0.

Key Points

1. A real number α is a root of the quadratic equation ax² + bx + c = 0 if aα² + bα + c = 0.
2. The zeroes of the quadratic polynomial ax² + bx + c and the roots of the quadratic equation are the same.
3. If we can factorise ax² + bx + c into a product of two linear factors, the roots can be found by equating each factor to zero.
4. Quadratic formula: The roots of a quadratic equation ax² + bx + c = 0 are given by x = -b ± √(b² - 4ac) / (2a), provided b² - 4ac is defined.
5. Nature of roots based on the discriminant D = b² - 4ac:
   • Two distinct real roots if D > 0
   • Two equal roots if D = 0
   • No real roots if D < 0

Examples

• Example of a quadratic equation: 2x² + x - 300 = 0.
• Example of finding roots by factorisation: 6x² - x - 2 = 0 can be factored to find roots.

Applications

• Quadratic equations arise in various real-life situations, such as calculating areas, dimensions, and solving problems involving products and sums.

Exercises

1. Find the nature of the roots of the following quadratic equations:
   • (i) 2x² - 3x + 5 = 0
   • (ii) 2x² - 6x + 3 = 0
2. Determine the values of k for quadratic equations to have two equal roots:
   • (i) 2x² + kx + 3 = 0
   • (ii) kx(x-2) + 6 = 0
3. Solve real-life problems involving quadratic equations, such as designing a rectangular park or finding ages based on given conditions.

Common Mistakes and Exam Tips for Quadratic Equations

Common Pitfalls

• Misidentifying Quadratic Equations: Ensure the equation is in the standard form ax² + bx + c = 0. Some equations may appear quadratic but are not.
• Ignoring the Discriminant: Always check the discriminant (b² - 4ac) to determine the nature of the roots. Misinterpretation can lead to incorrect conclusions about the existence of real roots.
• Negative Roots: When solving for dimensions (like length or breadth), remember that negative values are not valid in real-world contexts.
• Factorization Errors: Be cautious while factorizing quadratic equations. Incorrect factorization can lead to wrong roots.

Exam Tips

• Use the Quadratic Formula: When in doubt, apply the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a) to find roots accurately.
• Check Your Work: After finding roots, substitute them back into the original equation to verify correctness.
• Understand the Nature of Roots: Familiarize yourself with the conditions for the nature of roots based on the discriminant:
  • Two distinct real roots if b² - 4ac > 0
  • Two equal roots if b² - 4ac = 0
  • No real roots if b² - 4ac < 0
• Practice Factorization: Regularly practice factorization techniques to improve speed and accuracy in solving quadratic equations.