Equations and inequalities is a topic that appears across both papers of the Edexcel IGCSE (4MB1) Mathematics Specification B exam, and it sits at the heart of the algebra section. The ability to solve equations fluently is a prerequisite for everything from coordinate geometry to calculus.
Key facts
- This topic is among the most commonly examined areas in the specification.
- It covers linear equations, quadratic equations (by factorisation, completing the square, formula and graph), cubic equations, linear simultaneous equations, simultaneous equations where one is quadratic, linear inequalities in one and two variables, and quadratic inequalities.
- The edexcel igcse equations and inequalities content spans straightforward solving through to multi-step problems that combine equation types.
Linear equations
A linear equation contains no powers of x higher than 1. The aim is to isolate x on one side.
Worked example: Solve 5(2x - 3) = 3x + 12
Expand: 10x - 15 = 3x + 12
Subtract 3x: 7x - 15 = 12
Add 15: 7x = 27
Divide by 7: x = 27/7
The method is systematic: expand brackets, collect x terms on one side, collect constants on the other, divide. Every step should appear on a separate line.
Quadratic equations
A quadratic equation has the form ax2 + bx + c = 0. The specification requires four methods of solution, and the equations and inequalities explained here cover all four.
Method 1: Factorisation
Worked example: Solve 2x2 + 5x - 3 = 0
Find two numbers that multiply to 2 x (-3) = -6 and add to 5. Those numbers are 6 and -1.
Rewrite: 2x2 + 6x - x - 3 = 0
Group: 2x(x + 3) - 1(x + 3) = 0
(2x - 1)(x + 3) = 0
2x - 1 = 0, so x = 1/2
x + 3 = 0, so x = -3
Method 2: The quadratic formula
x = [-b +/- sqrt(b2 - 4ac)] / 2a
Worked example: Solve 3x2 - 7x + 2 = 0
a = 3, b = -7, c = 2
b2 - 4ac = 49 - 24 = 25
x = [7 +/- sqrt(25)] / 6 = [7 +/- 5] / 6
x = 12/6 = 2 or x = 2/6 = 1/3
Method 3: Completing the square
Worked example: Solve x2 + 6x - 4 = 0 by completing the square.
x2 + 6x = 4
(x + 3)2 - 9 = 4
(x + 3)2 = 13
x + 3 = +/-sqrt(13)
x = -3 + sqrt(13) or x = -3 - sqrt(13)
Completing the square also reveals the vertex of the parabola y = x2 + 6x - 4: the minimum point is at (-3, -13).
Method 4: Graphical solution
Plot y = ax2 + bx + c and read off where the curve crosses the x-axis. The x-coordinates of the crossings are the solutions. This method gives approximate answers, and the exam will usually specify the required degree of accuracy.
Cubic equations
The specification covers equations of third degree. These are typically solved by first finding one root using the factor theorem, then dividing the cubic by the corresponding linear factor to obtain a quadratic, which you solve normally.
Worked example: Solve x3 - 6x2 + 11x - 6 = 0
Try x = 1: 1 - 6 + 11 - 6 = 0. So (x - 1) is a factor.
Divide: x3 - 6x2 + 11x - 6 = (x - 1)(x2 - 5x + 6)
Factorise the quadratic: x2 - 5x + 6 = (x - 2)(x - 3)
Solutions: x = 1, x = 2, x = 3
Simultaneous equations: two linear
Solve by elimination or substitution. Elimination is generally faster when the coefficients align.
Worked example: Solve 3x + 2y = 14 and 5x - 2y = 2
Add the equations: 8x = 16, so x = 2
Substitute into the first: 6 + 2y = 14, so 2y = 8, y = 4
Solution: x = 2, y = 4
Check: 5(2) - 2(4) = 10 - 8 = 2. Correct.
Simultaneous equations: one linear, one quadratic
Use substitution: solve the linear equation for one variable, substitute into the quadratic, and solve the resulting quadratic.
Worked example: Solve y = x + 1 and y = x2 - 3x + 5
Substitute: x + 1 = x2 - 3x + 5
Rearrange: x2 - 4x + 4 = 0
(x - 2)2 = 0, so x = 2
y = 2 + 1 = 3
Solution: (2, 3). Since the quadratic has a repeated root, the line is tangent to the curve.
Linear inequalities
Linear inequalities are solved exactly like linear equations, with one critical exception: when you multiply or divide by a negative number, the inequality sign reverses.
Worked example: Solve 4 - 3x > 10
Subtract 4: -3x > 6
Divide by -3 (reverse the sign): x < -2
On a number line, x < -2 is shown with an open circle at -2 and an arrow pointing left. If the inequality were x <= -2, the circle would be filled (closed).
Inequalities in two dimensions
The what is equations and inequalities igcse specification also covers representing inequalities as regions on a coordinate plane. For y < 2x + 1, draw the line y = 2x + 1 as a dashed line and shade the region below it.
| Inequality symbol | Line style | Boundary included? |
|---|---|---|
| < or > | Dashed | No |
| <= or >= | Solid | Yes |
Quadratic inequalities
The edexcel igcse mathematics specification b definition requires you to solve quadratic inequalities in one variable and represent the solution set on a number line.
Worked example: Solve x2 - 5x + 6 < 0
Factorise: (x - 2)(x - 3) < 0
The roots are x = 2 and x = 3. The quadratic opens upward (positive coefficient of x2), so it is negative between the roots.
Solution: 2 < x < 3
On a number line: open circles at 2 and 3, with the region between them marked.
Worked example: Solve x2 + x - 12 >= 0
Factorise: (x + 4)(x - 3) >= 0
Roots: x = -4 and x = 3. The quadratic is non-negative outside and at the roots.
Solution: x <= -4 or x >= 3
Common mistakes
- Forgetting to reverse the inequality when dividing by a negative. This is the most frequent error in linear inequalities. Write the rule at the top of your working as a reminder.
- Only finding one solution to a quadratic. A quadratic has two solutions unless the discriminant is zero. Always check.
- Writing the solution to a quadratic inequality as a single range when it should be two separate regions. x2 > 9 gives x < -3 or x > 3, not -3 < x < 3.
- Confusing substitution and elimination in simultaneous equations. For two linear equations, elimination is usually cleaner. For one linear and one quadratic, substitution is the required method.
Self-check questions
- Solve 4(x - 2) + 3 = 2x + 7
Answer: 4x - 8 + 3 = 2x + 7, so 4x - 5 = 2x + 7, 2x = 12, x = 6. - Solve x2 - x - 20 = 0
Answer: (x - 5)(x + 4) = 0, x = 5 or x = -4. - Solve 2x + y = 7 and x2 + y = 10
Answer: y = 7 - 2x. Substitute: x2 + 7 - 2x = 10, x2 - 2x - 3 = 0, (x - 3)(x + 1) = 0. x = 3, y = 1 or x = -1, y = 9. - Solve 3 - 2x >= 9
Answer: -2x >= 6, x <= -3. - Solve x2 - 4x - 5 > 0
Answer: (x - 5)(x + 1) > 0. Roots at x = 5 and x = -1. Since the parabola opens upward, the inequality holds outside the roots: x < -1 or x > 5.
Exam approach: choosing the right method
The first 10 seconds of any equation question should be spent identifying what type of equation you are dealing with. Is it linear (no squared term)? Quadratic (highest power is 2)? Cubic (highest power is 3)? Are there two equations (simultaneous)? Is there an inequality sign instead of an equals sign? Once you have answered these questions, the method is determined, and you can proceed with confidence rather than guessing.
For quadratics, try factorisation first. If the numbers do not factorise neatly within 30 seconds, switch to the formula. Completing the square is typically only required when the question explicitly asks for it, or when you need to find the turning point of a parabola.
The edexcel igcse mathematics specification b notes on equations and inequalities span a wide range of question types, but every question reduces to a clear method if you identify the equation type first. Linear, quadratic or cubic? One equation or two? Equality or inequality? Answer those questions and the method follows. Practise on the Green Bridge CBT platform with edexcel igcse mathematics specification b explained problems sorted by difficulty, and track which equation types still cause you trouble so you can target them before the exam.
Edexcel IGCSE equations and inequalities explained: linear, quadratic and cubic equations, simultaneous equations, and solving inequalities with worked examples.
Àsìkò méjì (Comment(s))