Algebra for Edexcel IGCSE Mathematics Specification B: Methods and Worked Examples

Algebra is the structural backbone of the Edexcel IGCSE Mathematics Specification B specification. It permeates every section of the exam, and the topics within the Algebra chapter are among the most heavily examined in both papers.

The edexcel igcse mathematics specification b algebra content is divided into three areas: Algebraic Processes, Equations and Inequalities, and Sequences. Together, they form a coherent progression from manipulation to problem-solving. Algebraic fluency here is not optional; it is the prerequisite for success in functions, calculus, matrices, and coordinate geometry.

Algebraic processes

Collecting like terms and expanding brackets

These are foundational operations. Collecting like terms requires identifying terms with the same variable and power. Expanding brackets applies the distributive law.

Worked example: Expand and simplify (3x + 2)(2x - 5).

= 3x(2x) + 3x(-5) + 2(2x) + 2(-5)

= 6x² - 15x + 4x - 10

= 6x² - 11x - 10

Factorising

Factorisation is the reverse of expansion. The specification requires factorisation of simple algebraic expressions, including quadratics and expressions with common factors.

Worked example: Factorise 6x² + 7x - 20.

We need two numbers that multiply to give 6 x (-20) = -120 and add to give 7. Those numbers are 15 and -8.

Rewrite: 6x² + 15x - 8x - 20.

Group: 3x(2x + 5) - 4(2x + 5).

Factor: (3x - 4)(2x + 5).

The factor theorem

The factor theorem states: if f(a) = 0, then (x - a) is a factor of f(x). This extends to factors of the form (ax + b): if f(-b/a) = 0, then (ax + b) is a factor. The algebra edexcel igcse exam frequently tests this with cubic polynomials.

Worked example: Show that (x - 2) is a factor of f(x) = x³ - 3x² - 4x + 12, and hence factorise f(x) completely.

f(2) = 8 - 12 - 8 + 12 = 0. Since f(2) = 0, (x - 2) is a factor.

Now perform algebraic division:

x³ - 3x² - 4x + 12 divided by (x - 2):

x³ / x = x². Multiply: x²(x - 2) = x³ - 2x².

Subtract: -3x² - (-2x²) = -x². Bring down -4x: -x² - 4x.

-x² / x = -x. Multiply: -x(x - 2) = -x² + 2x.

Subtract: -4x - 2x = -6x. Bring down +12: -6x + 12.

-6x / x = -6. Multiply: -6(x - 2) = -6x + 12.

Subtract: 0.

So f(x) = (x - 2)(x² - x - 6) = (x - 2)(x - 3)(x + 2).

Algebraic division is a methodical process. Write each step clearly, aligning terms by degree. Errors almost always come from sign mistakes during the subtraction step. If your remainder is not zero when the factor theorem guarantees it should be, re-check your signs.

Algebraic fractions

The specification requires manipulation of algebraic fractions with numerical, linear, or quadratic denominators, including addition, subtraction, multiplication, and division.

Worked example: Simplify 2/(x + 3) + 3/(x - 1).

Common denominator: (x + 3)(x - 1).

= 2(x - 1)/((x + 3)(x - 1)) + 3(x + 3)/((x + 3)(x - 1))

= (2x - 2 + 3x + 9) / ((x + 3)(x - 1))

= (5x + 7) / ((x + 3)(x - 1))

Changing the subject of a formula

Worked example: Make r the subject of V = (4/3)pi r³.

Multiply both sides by 3: 3V = 4pi r³.

Divide by 4pi: r³ = 3V / (4pi).

Take the cube root: r = the cube root of (3V / (4pi)).

Equations and inequalities

Solving quadratic equations

The edexcel igcse mathematics specification b revision notes for this topic must cover all four methods: factorisation, the quadratic formula, completing the square, and graphical solution.

By factorisation: Solve x² - 5x + 6 = 0.

(x - 2)(x - 3) = 0, so x = 2 or x = 3.

By the quadratic formula: Solve 2x² + 3x - 7 = 0.

x = (-3 +/- sqrt(9 + 56)) / 4 = (-3 +/- sqrt(65)) / 4.

x = (-3 + sqrt(65))/4 or x = (-3 - sqrt(65))/4.

x = 1.27 or x = -2.77 (to 3 s.f.).

By completing the square: Solve x² + 6x - 2 = 0.

x² + 6x = 2.

(x + 3)² - 9 = 2.

(x + 3)² = 11.

x + 3 = +/- sqrt(11).

x = -3 + sqrt(11) or x = -3 - sqrt(11).

Simultaneous equations

The specification covers two types: two linear equations, and one linear with one quadratic.

Worked example (linear and quadratic): Solve y = x + 1 and y = x² - 3x + 5 simultaneously.

Substitute: x + 1 = x² - 3x + 5.

Rearrange: x² - 4x + 4 = 0.

(x - 2)² = 0, so x = 2.

y = 2 + 1 = 3.

Solution: x = 2, y = 3 (a repeated root, meaning the line is tangent to the curve).

Linear inequalities

Solving linear inequalities follows the same rules as equations, with one critical difference: multiplying or dividing by a negative number reverses the inequality sign.

Worked example: Solve 3 - 2x > 7.

-2x > 4.

x < -2 (sign reverses because we divide by -2).

On a number line, this is represented by an open circle at -2 with an arrow pointing left (toward smaller values). An open circle means -2 itself is not included.

Quadratic inequalities

The igcse 4mb1 algebra exam also tests quadratic inequalities. The method requires finding the critical values, then determining which region satisfies the inequality.

Worked example: Solve x² - 5x + 4 < 0.

Factorise: (x - 1)(x - 4) < 0.

Critical values: x = 1 and x = 4.

The quadratic opens upward (positive x² coefficient), so it is negative between the roots.

Solution: 1 < x < 4.

Quadratic inequality errors. The most common mistake is writing the solution as x < 1 or x < 4, treating each factor separately. A quadratic inequality requires you to consider the shape of the parabola. Sketch it: if the parabola opens upward, the expression is negative between the roots and positive outside them. If it opens downward, the reverse holds.

Inequalities in two dimensions

The specification requires representing linear inequalities in two-dimensional space. The line y = mx + c divides the plane into two regions. You shade the region that does not satisfy the inequality, leaving the solution region unshaded.

Sequences

The Sequences topic within the edexcel igcse mathematics specification b notes requires recognising and continuing sequences with a common difference (arithmetic sequences) or common integer patterns.

Worked example: Find the nth term of the sequence 5, 8, 11, 14, ...

Common difference d = 3.

The nth term of an arithmetic sequence is a + (n-1)d, where a is the first term.

nth term = 5 + 3(n - 1) = 5 + 3n - 3 = 3n + 2.

Check: When n = 1, 3(1) + 2 = 5. When n = 4, 3(4) + 2 = 14. Both match.

Common mistakes in Algebra

Sign errors in expansion and factorisation. (x - 3)² is x² - 6x + 9, not x² - 9. The middle term is commonly dropped.
Forgetting to reverse the inequality sign. When multiplying or dividing by a negative number, the inequality direction must flip. Missing this produces the opposite solution set.
Incomplete factorisation. After applying the factor theorem to a cubic, always check whether the resulting quadratic can be factorised further.
Algebraic fraction errors. When adding fractions with different denominators, every term in the numerator must be multiplied, not just the first. 2/(x+1) + 3/(x-2) requires multiplying all of 2 by (x-2) and all of 3 by (x+1).
Dropping the +/- in quadratic solutions. The quadratic formula and completing the square both produce two solutions. Missing one means half the answer is absent.

Self-check questions

Test yourself with these edexcel igcse mathematics specification b practice questions. Complete each one without looking at the solution first.

Factorise completely: 2x³ + x² - 13x + 6, given that (x + 3) is a factor.
Solve the simultaneous equations: y = 2x - 1 and x² + y² = 10.
Solve x² - 2x - 8 >= 0 and represent the solution on a number line.
Simplify: (x² - 9)/(x² + 5x + 6).

Solutions

1. Verify: f(-3) = 2(-27) + 9 + 39 + 6 = 0. Divide by (x + 3): 2x³ + x² - 13x + 6 = (x + 3)(2x² - 5x + 2) = (x + 3)(2x - 1)(x - 2).

2. Substitute y = 2x - 1 into x² + y² = 10: x² + (2x - 1)² = 10. x² + 4x² - 4x + 1 = 10. 5x² - 4x - 9 = 0. (5x - 9)(x + 1) = 0. x = 9/5 or x = -1. When x = 9/5, y = 13/5. When x = -1, y = -3. Solutions: (9/5, 13/5) and (-1, -3).

3. Factorise: (x - 4)(x + 2) >= 0. Critical values: x = 4 and x = -2. Parabola opens upward, so positive outside the roots. Solution: x <= -2 or x >= 4. On the number line, closed circles at -2 and 4 with arrows pointing outward.

4. Numerator: x² - 9 = (x - 3)(x + 3). Denominator: x² + 5x + 6 = (x + 2)(x + 3). Cancel (x + 3): (x - 3)/(x + 2), where x is not equal to -3.

The edexcel igcse mathematics specification b explained approach to Algebra emphasises precision and method. Every step in a factorisation, every line of an equation, every boundary on an inequality must be justified. The rigour required is what makes this section challenging, but it is also what makes it rewarding: a student who masters algebraic processes has the tools to tackle every other section of the specification with confidence.