Algebra, equations and sequences
Move from symbolic rules to solutions you can check by substitution.
What you will learn
- Expand, simplify and solve equations.
- Find a quadratic sequence rule and use composite functions.
Treat expressions and equations differently
An expression can be simplified; an equation can be solved. Distribute multiplication over every term inside a bracket, including negative signs. Collect only like terms. For simultaneous equations, substitution or elimination reduces two unknowns to one; substitute the result back into both equations.
For quadratics, factorise where possible and use the zero-product rule. A product is zero only if at least one factor is zero. Give every requested root, and check roots in the original equation.
Indices and inequalities
When dividing powers with the same non-zero base, subtract exponents. A negative index represents a reciprocal. When solving an inequality, multiplying or dividing both sides by a negative number reverses the inequality sign. Adding or subtracting does not reverse it.
Sequences and functions
For a quadratic sequence with consecutive integer indices, a constant second difference of 2a identifies the coefficient a in an²+bn+c. Subtract an² from the terms and find the remaining linear rule. Check several terms.
In f(g(x)), evaluate g first and then use its output as the input to f. Brackets are especially important when squaring negative numbers.
Worked example
Solve 2x² + x − 6 = 0.
Show the worked solution
- Look for factors whose product is −12 and sum is 1: 4 and −3.
- Write 2x² + 4x − 3x − 6 = 2x(x+2) − 3(x+2).
- Factorise as (2x−3)(x+2)=0 and solve each factor.
Answer x = 3/2 or x = −2
Common mistakes
- Do not cancel terms across addition.
- Reversing an inequality is required when dividing by a negative number.
- A sequence rule needs a defined starting index.
What can you explain now?
The quadratic sequence begins 2, 6, 12, 20 for n=1,2,3,4. Find its nth term.
Compare with the explanation
n²+n
The second difference is 2, suggesting n². Subtracting n² leaves 1,2,3,4, which is n.
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Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs contains 21; Coordinate geometry contains 10 teaching lessons and a mixed checkpoint. Nine earlier overviews support selected topics across the syllabus. The other six syllabus areas, full cumulative assessment and human teacher review are not yet complete. Written lesson practice is self-checked, not automatically graded. The checkpoint samples skills and does not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources