Maths 168 - Module 4

Module 4: Algebra II

Contents

• Expanding and Factorising Quadratic Expressions

• Factorising Quadratics

• Solving Quadratic Equations

• Simultaneous Equations

Expanding Expressions

• Expanding takes expressions with brackets and rewrites them without brackets.

  • For a single term outside brackets, multiply every term inside by that term.

    • Example: 2x(p^2 + 3q) = 2xp^2 + 6xq

• To expand two sets of brackets, multiply each term in one bracket by every term in the other.

  • Example: (2x + 1)(3x - 2) = 6x^2 + 2x - 3x - 2 = 6x^2 - x - 2

• Key Note: A negative outside switches signs inside brackets.

Factorising Expressions

• Factorising is reversing expanding: rewriting expressions without brackets into expressions with brackets.

• Common Factors:

  1. Find the highest common factor (HCF).

  2. Place HCF outside brackets.

  3. Divide terms by HCF to find what goes inside brackets.

  • Example: 8xy + 12y = 4y(2x + 3)

• Grouping: When terms don't share common factors, group terms with common factors together.

  • Example: 4ab + 4bc + 3ad + 3cd = 4b(a + c) + 3d(a + c) = (a + c)(4b + 3d)

Quadratic Expressions

• General form: ax² + bx + c where a ≠ 0.

• Factorising when a = 1: Find numbers that multiply to c and add to b.

  • Example: x² + 5x + 6 → (x + 2)(x + 3)

• Difference of Two Squares: a² - b² = (a - b)(a + b).

• Perfect Squares: Expressions like x² + 4x + 4 = (x + 2)².

Solving Quadratic Equations

• General form: ax² + bx + c = 0; a ≠ 0.

• Methods:

  1. Factorisation (e.g., x² + 5x + 6 = 0 → (x + 2)(x + 3) = 0).

  2. Quadratic Formula: x = -b ± √D / 2a, where D = b² - 4ac.

  • Example of Quadratic Formula:

    • For x² - 3x - 5 = 0, identify a = 1, b = -3, c = -5, then D = 9 + 20 = 29.

• Application Examples:

  • Find two consecutive numbers whose product is 72.

  • A rectangular hall dimensions based on area given.

Simultaneous Equations

• Set of equations to be solved together, typically linear.

• Methods include substitution, elimination, and using matrices.

  • Example equations:

    • x + y = 3

    • 3x + y = 4

• Elimination Method: Align and manipulate equations to eliminate variables by addition or subtraction.

• Example problems focus on applying learned techniques to solve for multiple variables.