quadratic

Quadratic Equations Overview

  • Definition: Quadratic equations are polynomial equations of the form ax² + bx + c = 0, where a ≠ 0.

  • Methods of Solution:

    • Factorisation

    • Completing the Square

    • Using the Quadratic Formula

    • Using Graphs

  • Practice: Mastery requires frequent practice with a variety of exercises.

Content Structure

  1. Introduction

  2. Solving quadratic equations by factorisation

  3. Solving quadratic equations by completing the square

  4. Solving quadratic equations using a formula

  5. Solving quadratic equations by using graphs

1. Introduction

  • Key Points:

    • Quadratic equations have a term involving x²

    • May include terms with x (e.g., 5x)

    • May have constant terms (e.g., 6, -7, etc.)

    • Cannot have terms with higher powers or negative powers of x.

  • Standard Form: ax² + bx + c = 0, where a ≠ 0.

  • Parameter Descriptions:

    • a: coefficient of x², must be non-zero.

    • b: coefficient of x.

    • c: constant term.

2. Solving Quadratic Equations by Factorisation

  • Prerequisite Knowledge: Understanding how to factorise quadratic expressions.

  • Example Solutions:

    • Example 1: Solve 3x² = 27.

      • Rewrite: 3x² - 27 = 0 → 3(x² - 9) = 0 → 3(x - 3)(x + 3) = 0

      • Solutions: x = 3, x = -3

    • Example 2: Solve 5x² + 3x = 0.

      • Factor: x(5x + 3) = 0 → Solutions: x = 0, x = -3/5.

      • Note: Avoid canceling the common factor x too early to preserve all solutions.

3. Completing the Square

  • Method: Rewrite the quadratic to isolate a perfect square.

    • Example Solution: Solve x² - 3x - 2 = 0.

    • Rewrite: (x - 3/2)² - (3/2)² - 2 = 0

    • Simplify: (x - 3/2)² - 17/4 = 0 → (x - 3/2)² = 17/4

    • Solutions: x = (3 ± √17)/2.

    • Conclusion: Solutions can be approximated using a calculator.

4. Using the Quadratic Formula

  • Formula: For ax² + bx + c = 0, the solutions are given by x = -b ± √(b² - 4ac) / 2a.

  • Importance: It's essential to memorize the formula.

  • Example: Solve x² - 3x - 2 = 0.

    • Identify: a = 1, b = -3, c = -2.

    • Substitute: x = -(-3) ± √((-3)² - 4 × 1 × (-2)) / (2 × 1)

    • Calculate: x = 3 ± √17 / 2.

  • Another Example: Solve 3x² = 5x - 1 → 3x² - 5x + 1 = 0

    • Identify values and apply the quadratic formula.

5. Using Graphs to Solve Quadratics

  • Graph Shape: The graph of y = ax² + bx + c depends on the sign of a.

    • If a > 0, the graph opens upward.

    • If a < 0, the graph opens downward.

  • Solution Finding: Points where the graph intersects the x-axis correspond to solutions for y = 0.

    • Examples:

      • Solution of x² - 3x - 2 = 0 involves observing graph behavior crossing the x-axis to find roots.

      • Determine x-intercepts by plotting values and analyzing the graph structure.

    • Conclusion: Graphs provide a visual means of identifying real roots and can also be used for additional equations by analyzing horizontal lines.

Exercises

  • Factorisation: Solve various quadratic equations by factorising.

  • Completing the Square: Use this method to solve given equations.

  • Using the Formula: Apply the quadratic formula to specified equations.

  • Graphing: Plot graphs and use them to find solutions to quadratic equations, providing approximate answers when necessary.