This page gives you the key ideas to recognise before you start a question. The full course then takes you through the method, exam-style practice and fully worked solutions.
Choose the method
Factorise when the structure is friendly; use the quadratic formula for a reliable general method; complete the square when the form of the graph or turning point matters.
Use the discriminant
For ax² + bx + c = 0, the sign of b² − 4ac tells you whether there are two distinct real roots, one repeated real root, or no real roots.
Connect algebra and graphs
Roots are x-intercepts. Completing the square exposes the turning point and helps with sketching and range questions.
Check your solution
Substitute roots back when useful and make sure your answer fits the question, especially in modelling contexts.
Worked example: use the discriminant
For x² − 6x + k = 0 to have one repeated root:
b² − 4ac = 0
(−6)² − 4(1)(k) = 0
36 − 4k = 0
k = 9
b² − 4ac = 0
(−6)² − 4(1)(k) = 0
36 − 4k = 0
k = 9
Common mistakes to avoid
- Using the discriminant but forgetting that a repeated root means it equals 0.
- Losing the sign of b when substituting into the quadratic formula.
- Giving algebraic roots without checking whether a modelling context restricts them.