Skill paths

Use a familiar equation, then add a parameter and test your reasoning under pressure.

1. 1

   Solve a GCSE quadratic

   Factorise and check both roots.

   Edexcel Higher →AQA Higher →
2. 2

   Handle a repeated root

   Use the discriminant when a number in the equation can change.

   OCR A-level →Edexcel A-level →
3. 3

   Try the TMUA version

   Choose the number of parameter values, then explain why.

   TMUA-style question →

Move from solving intervals to deciding when a statement is both necessary and sufficient.

1. 1

   Find the GCSE interval

   Remember to include the negative side of a squared inequality.

   Edexcel Higher →AQA Higher →
2. 2

   Solve a quadratic inequality

   Use the roots to decide which interval works.

   OCR A-level →Edexcel A-level →
3. 3

   Test both directions

   See whether the interval is exactly equivalent to the original condition.

   TMUA-style question →

Start with a GCSE factorisation and use it to count increasingly subtle divisors.

1. 1

   Find the prime factors

   Write a number as a product of prime powers.

   Edexcel Higher →AQA Higher →
2. 2

   Count square divisors

   Notice what happens to exponents when the divisor is a square.

   SMC-style question →
3. 3

   Take the AIME step

   Apply the same idea to a larger product without listing every divisor.

   AIME-style question →