Algebra readiness
Learn one idea at a time
Choose a unit. Each has worked examples and practice.
Read quadratic roots and signs
Recognise number systems, complete a square and connect roots with strict signs.
Factor polynomials and simplify surds
Use the factor theorem, algebraic identities and valid radical operations.
Work with indices and exponential equations
Keep domains visible and choose a common base or a positive substitution.
Translate and solve with logarithms
Apply logarithm laws with their base and argument restrictions.
Choose and solve partial fractions
Divide when needed, factor the denominator and retain every required numerator and power.
Use radians and triangle ratios
Match angles with their sides and distinguish right triangles from general triangles.
Read sine, cosine and tangent graphs
Connect domains, ranges, periods, symmetry and excluded tangent inputs.
Choose principal values and valid signs
Use principal ranges and check identity denominators and square-root signs.
Use compound angles and R-formulae
Retain angle signs, derive equivalent forms and use amplitude to bound an expression.
Differentiate sums and compositions
Choose the appropriate rule and retain the elementary derivative conditions.
Integrate as reverse differentiation
Recognise standard antiderivatives and account for a linear inner function.
Choose the next readiness practice
Use the retained tutorial to check the skills you have revisited.
By the end, you can
- Refresh equations, powers and logarithms
- Complete the square and use the factor theorem
- Recognise partial fractions, trigonometry and basic calculus
12 focused units · about an hour for the full chapter. Allow extra time for practice.
The central idea
Equivalent algebraic forms reveal different information. Factoring finds zeros, completing the square exposes a quadratic vertex, and logarithms turn multiplication into addition. Keep the number system and domain in view: denominators cannot vanish and real logarithms need positive arguments.
Worked example
Solve . Completing the square gives , hence or . The same expression is ; its vertex form also shows that its minimum is at .
Squaring can introduce candidates that do not satisfy the original equation.
Check your understanding
Solve over the real numbers.
The equation requires . Squaring gives , so candidates are 2 and −1. Only satisfies both the domain and the original equation.
Practice questions
47 examples and practice questions, in 8 sections.
More practice
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