Algebra – Equations and Inequalities (Cambridge IGCSE 0580)
This set of notes follows the 2025‑27 Cambridge IGCSE Mathematics syllabus (C2.1‑C2.11, E2.5‑E2.7). It covers the required algebraic techniques, shows how they link to other strands (sequences, graphs, practical situations) and highlights the assessment objectives (AO1 knowledge; AO2 analysis & communication).
1. Introduction to Algebra (C2.1)
- Letters such as a, b, x, y represent numbers. They can stand for any real number unless a condition (e.g. a≠0) is given.
- Substitution: replace a letter by its value and evaluate.
Example: If a = 4, evaluate 3a + 2 → 3·4 + 2 = 14.
2. Algebraic Manipulation (C2.2)
2.1 Simplifying & Expanding
- Combine like terms: 3x + 5x = 8x.
- Use the distributive law: a(b + c) = ab + ac.
Example: 2(3x – 4) = 6x – 8.
2.2 Factorising
- Common factor: ax + bx = (a + b)x.
- Factorising a quadratic (ax² + bx + c) – look for two numbers that multiply to ac and add to b.
Example: x² – 5x + 6 = (x – 2)(x – 3).
- Difference of squares: a² – b² = (a – b)(a + b).
2.3 Checking your work (AO2)
After simplifying, expanding or factorising, substitute a convenient value for the variable to verify that both sides are equal.
3. Linear Equations (C2.4)
3.1 Standard form
A linear equation in one variable can be written as
$$ax + b = 0 \qquad (a eq 0)$$
3.2 Solving steps
- Isolate the term containing the variable (move constants to the other side).
- Divide by the coefficient of the variable.
- Check by substituting the solution back into the original equation.
3.3 Example – whole numbers
$$3x - 7 = 2 \;\Longrightarrow\; 3x = 9 \;\Longrightarrow\; x = 3$$
Check: \(3(3)-7 = 2\) ✓
3.4 Example – fractions
$$\frac{2}{3}x - 5 = \frac{1}{4} \;\Longrightarrow\; \frac{2}{3}x = 5 + \frac{1}{4}
= \frac{21}{4} \;\Longrightarrow\; x = \frac{21}{4}\cdot\frac{3}{2}= \frac{63}{8}=7.875$$
Multiplying by the LCD (12) first avoids fractions.
3.5 Changing the subject of a formula (C2.5)
Isolate the required variable using the same steps as for a linear equation.
Example: From \(A=\frac12 bh\) express \(b\) in terms of \(A\) and \(h\).
$$A=\frac12 bh \;\Longrightarrow\; b = \frac{2A}{h}$$
4. Linear Inequalities (C2.6)
4.1 Single‑variable inequalities
- Same operations as for equations, but reverse the inequality sign when multiplying or dividing by a negative number.
4.2 Example
$$4 - 2x > 10 \;\Longrightarrow\; -2x > 6 \;\Longrightarrow\; x < -3$$
Solution set: \(\displaystyle x\in(-\infty,-3)\).
4.3 Representing on a number line
- Open circle at \(-3\) (strict inequality) and shade to the left.
4.4 Two‑variable linear inequalities (extended C2.6)
- Write the boundary line \(ax+by=c\). Use a solid line for ≤ or ≥, a dashed line** for < or >.
- Pick a test point (usually the origin) and see which side satisfies the inequality.
- Shade the entire half‑plane that satisfies the inequality.
4.5 Example – sketch \(2x - y \le 4\)
- Boundary: \(2x - y = 4 \;\Longrightarrow\; y = 2x - 4\) (solid line).
- Test (0,0): \(0 \le 4\) true → shade the side containing the origin.
5. Simultaneous Linear Equations (C2.5)
5.1 Methods
| Method | Key steps |
|---|
| Substitution |
- Solve one equation for a variable.
- Substitute that expression into the other equation.
- Solve the resulting single‑variable equation.
- Back‑substitute to obtain the second variable.
|
| Elimination (addition) |
- Make the coefficients of one variable opposites (multiply if necessary).
- Add the equations to eliminate that variable.
- Solve for the remaining variable.
- Substitute back to find the other variable.
|
5.2 Example – Elimination
\[
\begin{cases}
2x + 3y = 12\\[2pt]
4x - y = 5
\end{cases}
\]
- Multiply the second equation by 3: \(12x - 3y = 15\).
- Add to the first: \(14x = 27 \;\Longrightarrow\; x = \dfrac{27}{14}\).
- Substitute into \(4x - y = 5\): \(4\left(\dfrac{27}{14}\right) - y = 5\) → \(y = \dfrac{19}{7}\).
Solution: \(\displaystyle\left(\frac{27}{14},\frac{19}{7}\right)\). Check both equations – they hold.
5.3 Solving three‑variable systems (extended C2.5)
Use elimination twice to reduce to two equations, then apply any of the two‑variable methods. The same principles of checking apply.
6. Quadratic Equations (E2.5 – E2.7)
6.1 Standard form and discriminant
$$ax^{2}+bx+c=0\qquad(aeq0)$$
Discriminant \(\displaystyle\Delta = b^{2}-4ac\) determines the nature of the roots:
| \(\Delta\) | Roots |
|---|
| \(\Delta>0\) | Two distinct real roots |
| \(\Delta=0\) | One repeated real root |
| \(\Delta<0\) | Two complex conjugate roots (not required for IGCSE core) |
6.2 Solution techniques (always applicable)
- Factorising (when possible)
- Write \(ax^{2}+bx+c\) as a product of two linear factors.
- Set each factor to zero and solve.
Example: \(x^{2}-5x+6=0\) → \((x-2)(x-3)=0\) → \(x=2\) or \(x=3\).
- Completing the square (useful when \(a=1\) or after dividing by \(a\))
- Rewrite \(x^{2}+bx\) as \(x^{2}+bx+\left(\frac{b}{2}\right)^{2}-\left(\frac{b}{2}\right)^{2}\).
- Group to obtain a perfect square on the left.
- Take square roots and solve for \(x\).
Example: \(x^{2}+6x+5=0\)
\[
\begin{aligned}
x^{2}+6x &= -5\\
x^{2}+6x+9 &= 4\\
(x+3)^{2} &= 4\\
x+3 &= \pm2\\
x &= -1 \text{ or } -5
\end{aligned}
\]
- Quadratic formula (works for any quadratic)
\[
x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}
\]
Example: \(2x^{2}-4x-6=0\)
\[
x=\frac{4\pm\sqrt{16+48}}{4}
=\frac{4\pm8}{4}
\Longrightarrow x=3\text{ or }x=-1
\]
6.3 Vertex, axis of symmetry and graph (E2.7, C2.11)
For \(y=ax^{2}+bx+c\):
- Axis of symmetry: \(x = -\dfrac{b}{2a}\).
- Vertex \((h,k)\):
\[
h=-\frac{b}{2a},\qquad
k=c-\frac{b^{2}}{4a}\;( \text{or }k=a h^{2}+b h +c )
\]
- The parabola opens upwards if \(a>0\) and downwards if \(a<0\).
6.4 Example – Vertex of \(y=2x^{2}-8x+3\)
\[
h=-\frac{-8}{2\cdot2}=2,\qquad
k=2(2)^{2}-8(2)+3=8-16+3=-5
\]
Vertex \((2,-5)\); opens upwards because \(a=2>0\).
7. Sequences (C2.7)
7.1 Linear (arithmetic) sequences
General term: \(\displaystyle T_{n}=a+(n-1)d\) where \(a\) = first term, \(d\) = common difference.
Example: 4, 9, 14, … → \(a=4,\;d=5\) → \(T_{n}=4+5(n-1)=5n-1\).
7.2 Quadratic sequences
Second differences are constant. General term: \(\displaystyle T_{n}=an^{2}+bn+c\).
Example: 2, 6, 12, 20, …
- First differences: 4, 6, 8
- Second differences: 2 (constant) → quadratic.
- Assume \(T_{n}=an^{2}+bn+c\). Using \(n=1,2,3\):
\[
\begin{cases}
a+b+c = 2\\
4a+2b+c = 6\\
9a+3b+c = 12
\end{cases}
\;\Longrightarrow\; a=1,\;b=1,\;c=0
\]
- Hence \(T_{n}=n^{2}+n\).
8. Graphs in Practical Situations (C2.9)
8.1 Distance‑time graph
- A straight line through the origin represents constant speed.
- Gradient = speed (units m s⁻¹).
- Area under the line (if required) gives distance.
8.2 Conversion graphs
Example: miles ↔ kilometres (1 mi ≈ 1.609 km).
- Plot (0,0) and a second point, e.g. (5 mi, 8.045 km).
- Draw the straight line; its gradient is the conversion factor 1.609.
- Read any point on the line to obtain an instant conversion.
9. Graphs of Functions (C2.10)
| Function type | Typical equation | Key features |
|---|
| Linear | \(y = ax + b\) | Slope \(a\); y‑intercept \(b\); straight line. |
| Quadratic | \(y = ax^{2}+bx+c\) | Parabola; vertex \((h,k)\); axis \(x=h\); opens up if \(a>0\) else down. |
| Reciprocal | \(y = \dfrac{k}{x}\) | Two hyperbolic branches; vertical asymptote \(x=0\); horizontal asymptote \(y=0\). |
| Exponential | \(y = a\,b^{x}\;(b>0,\;beq1)\) | Growth if \(b>1\); decay if \(0
|
9.1 Example – Sketching \(y=\dfrac{2}{x}\)
- Mark points \((1,2),\;(-1,-2),\;(2,1),\;(-2,-1)\).
- Draw smooth curves approaching but never touching the axes.
9.2 Example – Exponential growth \(y=3\cdot2^{x}\)
- y‑intercept at \((0,3)\).
- Each unit increase in \(x\) doubles the value: \((1,6),\;(2,12),\;(3,24)\).
- Sketch a smooth curve rising rapidly to the right, approaching the x‑axis on the left.
10. Sketching Curves (C2.11)
10.1 Linear graphs
- Find y‑intercept \(b\) and gradient \(a\).
- Plot the intercept and a second point using the gradient.
- Draw a straight line through the points and extend.
10.2 Quadratic graphs
- Calculate the discriminant \(\Delta\) to know the number of real x‑intercepts.
- If \(\Delta\ge0\), find the roots (x‑intercepts) by factorising, completing the square, or using the formula.
- Find the vertex \((h,k)\) using \(h=-\dfrac{b}{2a}\) and \(k=c-\dfrac{b^{2}}{4a}\).
- Draw the axis of symmetry \(x=h\). Plot the vertex and any intercepts, then sketch a smooth parabola.
10.3 Example – Sketch \(y = -x^{2}+4x-3\)
- a = –1 (opens downwards).
- \(\Delta = 4^{2}-4(-1)(-3)=16-12=4>0\) → two real roots.
- Roots: \(x=\dfrac{-4\pm\sqrt{4}}{-2}=1\) and \(3\).
- Vertex: \(h=-\dfrac{4}{2(-1)}=2,\;k= -2^{2}+4(2)-3=1\).
- Plot points (1,0), (3,0), (2,1) and draw a downward‑opening parabola.
11. Quick Reference Table (AO1 Summary)
| Topic | Key formula / rule | Typical exam task |
|---|
| Linear equation | \(ax+b=0\;\Rightarrow\;x=-\dfrac{b}{a}\) | Solve, change the subject, check |
| Linear inequality | Reverse sign when multiplying/dividing by negative | Solve, represent on number line, sketch half‑plane |
| Simultaneous equations | Substitution or elimination | Find ordered pair, verify |
| Quadratic equation | \(ax^{2}+bx+c=0\); \(\Delta=b^{2}-4ac\) | Factorise, complete square, or use formula |
| Vertex of parabola | \(h=-\dfrac{b}{2a},\;k=c-\dfrac{b^{2}}{4a}\) | Find vertex, axis, sketch |
| Arithmetic sequence | \(T_{n}=a+(n-1)d\) | Find nth term or sum |
| Quadratic sequence | \(T_{n}=an^{2}+bn+c\) | Determine formula from first three terms |
| Practical graphs | Gradient = rate, intercept = constant | Interpret distance‑time, conversion, etc. |
12. Tips for the Exam (AO2)
- Read the question carefully – note whether a solution set, a graph or a formula is required.
- Show every step: marks are awarded for the method as well as the final answer.
- Always check your answer by substitution (equations) or by testing a point (inequalities).
- When working with fractions, clear denominators first to avoid arithmetic errors.
- For quadratics, a quick discriminant check tells you whether you need to find real roots or just state “no real solution”.
- Label axes, vertices, intercepts and asymptotes clearly on all sketches.