Topic 1: Chapter 8 — Algebraic fractions, quadratic expressions and equations

Expanding expressions

Involves multiplying out terms.

Distributive Law

a(b + c) = ab + ac; a(b - c) = ab - ac; (a + b)(c + d) = ac + ad + bc + bd.

Perfect Squares

(a + b)² = a² + 2ab + b²; (a - b)² = a² - 2ab + b².

Difference of Two Squares

(a + b)(a - b) = a² - b².

Factorising expressions

Reverse of expanding.

Common factor

Taking out the highest common factor to factorise.

Grouping

Factorising four-term expressions by grouping terms.

Factorising monic quadratics

Factorising quadratic trinomials with leading coefficient 1.

Factorising non-monic quadratics

Factorising quadratic trinomials with leading coefficient ≠ 1.

Standard quadratic form

ax² + bx + c.

Completing the square

Add (b/2)² to make perfect square: (x + b/2)².

Turning point form of quadratic

y = a(x - h)² + k.

Solving quadratic equations

By factorising, completing the square, or quadratic formula.

Quadratic formula

x = [-b ± √(b² - 4ac)] / 2a.

Discriminant

Δ = b² - 4ac; determines number of real solutions (Δ = 0 one solution, Δ < 0 no solution).

Null factor law

If A × B = 0, then A = 0 or B = 0.

Canceling common factors (algebraic fractions)

Divide top and bottom by the same expression

Simplify algebraic fractions (addition/subtraction)

Find a common denominator, then combine like terms

Multiply algebraic fractions

Multiply numerators and denominators separately, then simplify

Divide algebraic fractions

Multiply by the reciprocal of the second fraction

Solve equations with algebraic fractions

Multiply both sides by the lowest common denominator (LCD)

Perfect square trinomials

a² ± 2ab + b² = (a ± b)²

Difference of squares

a² - b² = (a - b)(a + b)

Quadratic formula

x = (-b ± √(b² - 4ac)) / 2a

Discriminant (Δ)

b² - 4ac, tells number of real roots

Use discriminant: 2 roots if

Δ > 0

Use discriminant: 1 root if

Δ = 0

Use discriminant: 0 roots if

Δ < 0

Worded quadratic problems

Convert scenario into equation (area, motion, etc.)

Solve quadratics by completing square

Rewrite as (x ± a)² = b, then solve

Expand (x + a)(x + b)

x² + (a + b)x + ab

Solve quadratic inequality

Solve related equation, test intervals on number line

Sketch quadratic from factorised form

y = a(x - p)(x - q) → roots: p, q; axis: midpoint of roots

Sketch quadratic from vertex form

y = a(x - h)² + k → vertex at (h, k)

Turning point formula (from standard form)

x = -b / 2a