Everything about a quadratic follows from its completed-square form. Once you can write it as a(x + p)² + q you can read off the vertex, the maximum or minimum, the line of symmetry and the range without drawing anything.
Syllabus points covered
- Complete the square and use it to find the vertex, line of symmetry and range
- Find the discriminant and use it to determine the number of real roots
- Solve quadratic equations by factorising, completing the square and the formula
- Solve quadratic inequalities and represent the solution correctly
- Solve equations in x that are quadratic in some function of x
Common mistakes
- Forgetting to factor the coefficient of x² out of the first two terms before completing the square.
- Writing the vertex as (p, q) when the form is a(x + p)² + q — the x-coordinate is −p.
- Giving a quadratic inequality's answer as a single chain like 2 < x < 5 when the parabola opens upward and the solution is actually x < 2 or x > 5. Sketch it.
- Losing the second solution when solving a disguised quadratic — remember to go back and solve for the original variable.
Worksheets
Use these as soon as you've learned the topic and need to practise it. Start with the worksheet, check yourself against the answers, and only then look at the step-by-step solutions.
- Completing the square
- Completing the square — answers
- The discriminant
- The discriminant — answers
- Quadratic inequalities
- Quadratic inequalities — answers
- Step-by-step solutionsComing soon
Topical past papers
Move on to these once you've worked through the worksheets and are ready for harder, exam-style questions. Real past-paper questions on this topic, with the official mark scheme and full worked solutions.
- Topical questionsComing soon
- Mark schemeComing soon
- Step-by-step worked solutionsComing soon
Looking for whole papers by session? All Additional Mathematics past papers