Algebraic Processes

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Algebraic processes form the backbone of WASSCE mathematics and appear in virtually every paper. The core skills you need are simplifying expressions, solving linear and quadratic equations, manipulating algebraic fractions, and expanding brackets using the distributive law. When simplifying, collect like terms — for example, 3x² + 5x − 2x² + 4 = x² + 5x + 4. When solving linear equations, isolate the unknown term; for 4x − 7 = 13, add 7 to both sides to get 4x = 20, then divide by 4 to find x = 5.

Essential Formulas:

• Quadratic formula: x = (−b ± √(b² − 4ac)) / 2a for ax² + bx + c = 0
• Difference of two squares: a² − b² = (a + b)(a − b)
• Perfect square trinomials: (a ± b)² = a² ± 2ab + b²
• Sum and product of roots: α + β = −b/a and αβ = c/a

Key Facts:

• Always check for common factors before applying any other factorisation technique
• When solving quadratic equations, first check whether the expression factorises neatly — try factors of ac that sum to b
• The discriminant Δ = b² − 4ac determines the nature of roots: Δ > 0 (two distinct real roots), Δ = 0 (equal roots), Δ < 0 (no real roots)
• Algebraic fractions require a common denominator; multiply through by the LCD to clear fractions

⚡ Exam Tip: In WASSCE, algebraic process questions frequently combine simplification with substitution. If a question asks you to “evaluate” an expression by substituting values, simplify algebraically FIRST before substituting — it reduces arithmetic errors. Watch out for questions combining factorisation with evaluation — common in Paper 2 Section A.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Algebraic Expressions and Manipulation

An algebraic expression combines numbers and variables using arithmetic operations. Terms are the separated parts connected by + or − signs. Like terms have the same variable part raised to the same power: 7xy² and −3xy² are like terms, but 7xy² and 7x²y are not. To simplify, collect like terms — this is the most fundamental algebraic skill tested at WASSCE level.

Factorisation Techniques

Factorisation is the reverse of expansion. The main methods tested at WASSCE are:

1. Common factor method: Extract the highest common factor (HCF) from all terms. Example: 6x²y + 9xy² = 3xy(2x + 3y)
2. Difference of two squares (DOTS): a² − b² = (a + b)(a − b). Example: x² − 16 = (x + 4)(x − 4)
3. Trinomial factorisation: For ax² + bx + c, find two numbers that multiply to ac and sum to b. Example: x² + 7x + 12 = (x + 3)(x + 4) because 3 × 4 = 12 and 3 + 4 = 7
4. Completing the square: Express ax² + bx + c in the form a(x − p)² + q

Solving Equations

Linear equations: Bring all terms containing the unknown to one side, constants to the other. Example: 5(2x − 3) − 3(x + 4) = 2 → 10x − 15 − 3x − 12 = 2 → 7x = 29 → x = 29/7

Quadratic equations: Three methods — factorisation, completing the square, and the quadratic formula. The quadratic formula always works: x = (−b ± √(b² − 4ac)) / 2a. For x² − 5x + 6 = 0, factorises to (x − 2)(x − 3) = 0, so x = 2 or x = 3.

Algebraic Fractions

Treat algebraic fractions the same way as numerical fractions. To add 1/(x + 2) + 3/(x − 1), the LCD is (x + 2)(x − 1). Rewrite: (x − 1 + 3x + 6) / ((x + 2)(x − 1)) = (4x + 5) / ((x + 2)(x − 1)). Always state restrictions: x ≠ −2 and x ≠ 1.

Inequalities

Solving inequalities follows the same steps as equations, with one critical difference: multiplying or dividing both sides by a negative number reverses the inequality sign. Example: −3x > 12 → dividing by −3 gives x < −4.

Problem-Solving Strategies:

• Read the problem twice before starting — identify what you need to find
• Translate word problems into algebraic expressions (e.g., “three more than a number” → x + 3)
• For word problems involving ages, sums, or quantities, assign a variable to the smallest quantity and express everything else in terms of it
• Always check your answer by substituting back into the original equation

Common Mistakes:

• Forgetting to multiply every term inside brackets when expanding — 2(x + 3) = 2x + 6, NOT 2x + 3
• Cancelling terms across addition/subtraction — you cannot cancel x in (x + 5)/x = (x/x) + (5/x); the correct simplification requires dividing each term
• Sign errors when moving terms across the equals sign — moving +7 to the other side gives −7, not +7
• Confusing factorisation with expansion — factorisation is going from expanded to factored form

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Derivation of the Quadratic Formula

Starting from ax² + bx + c = 0 (where a ≠ 0):

1. Divide by a: x² + (b/a)x + (c/a) = 0
2. Complete the square: x² + (b/a)x = −(c/a)
3. Add (b/2a)² to both sides: x² + (b/a)x + (b/2a)² = −(c/a) + (b/2a)²
4. Left side is a perfect square: (x + b/2a)² = (b² − 4ac) / 4a²
5. Take square root: x + b/2a = ±√(b² − 4ac) / 2a
6. Therefore: x = (−b ± √(b² − 4ac)) / 2a

The discriminant Δ = b² − 4ac determines the nature of the roots regardless of the value of a. This is crucial for determining whether a quadratic equation has real solutions before attempting to solve it.

Algebraic Identities — Proofs and Applications

Difference of cubes: a³ − b³ = (a − b)(a² + ab + b²) Sum of cubes: a³ + b³ = (a + b)(a² − ab + b²)

These are commonly tested in WASSCE Paper 1. Example: 64x³ − 27 = (4x)³ − 3³ = (4x − 3)(16x² + 12x + 9).

Simultaneous Equations

Two equations in two unknowns. Methods: substitution and elimination.

Example: 3x + 2y = 16 and 2x − y = 3 Multiply the second equation by 2: 4x − 2y = 6 Add to the first: 7x = 22 → x = 22/7 Substitute back: 2(22/7) − y = 3 → y = 44/7 − 3 = 23/7

For simultaneous equations where one is linear and one quadratic, substitute the linear expression into the quadratic.

Indices Laws (Brief Review)

• a^m × a^n = a^(m+n)
• a^m ÷ a^n = a^(m−n)
• (a^m)^n = a^(mn)
• a^0 = 1 (for a ≠ 0)
• a^(−n) = 1/a^n
• a^(m/n) = (ⁿ√a)^m = ⁿ√(a^m)

These laws are essential when simplifying algebraic expressions involving powers.

Word Problem Patterns

WASSCE frequently tests algebraic word problems in Paper 2 Section A. Common patterns include:

• Number problems: “The sum of three consecutive integers is 72. Find them.” → Let n, n+1, n+2 be the integers. n + n+1 + n+2 = 72 → 3n = 69 → n = 23. Numbers are 23, 24, 25.
• Age problems: “A father is four times as old as his son. In 20 years, he will be twice as old.” → Let son’s age = s. Father’s age = 4s. After 20 years: 4s + 20 = 2(s + 20) → 4s + 20 = 2s + 40 → 2s = 20 → s = 20. Father is 80.
• Distance/speed problems: Use the relationship distance = speed × time. Often involve two travellers moving in opposite directions or the same direction.
• Work/rate problems: If A can complete a task in x hours and B in y hours, together they take xy/(x+y) hours.

Exam Pattern Analysis

WASSCE Paper 1 (objective): 50 questions, 1 hour 30 minutes. Algebra questions typically constitute 15–20 questions, covering simplification, factorisation, substitution, and solving equations.

WASSCE Paper 2 (essay): 2 hours 30 minutes. Section A has 5 compulsory questions on Core Mathematics topics including Algebra. Section B has 4 optional questions. Algebra in Section A often involves solving quadratic equations, factorisation proofs, and word problems leading to quadratic equations.

⚡ Exam Strategy: When a quadratic equation word problem asks for “the value of x” and gives multiple conditions, form the equation from the conditions, then solve. Always reject any solution that makes physical sense impossible (e.g., negative length, or a fraction of a person).

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