Two equations, two unknowns — but now one of them can be a curve. Solving a linear equation together with a quadratic is really asking where a line meets a curve, and the discriminant of the equation you end up with tells you whether it cuts, touches, or misses entirely.
Syllabus points covered
- Solve simultaneous equations in two unknowns where one equation is linear and one is quadratic
- Interpret the solutions as the points of intersection of a line and a curve
- Use the discriminant to determine whether a line intersects, is tangent to, or does not meet a curve
- Find the condition for a line to be a tangent to a curve
Common mistakes
- Substituting the quadratic into the linear equation instead of the other way round. Always rearrange the linear one first.
- Finding the x-values and stopping. Each x needs its matching y, and they must be paired correctly.
- Squaring both sides carelessly and introducing solutions that don't satisfy the original equations. Always check them back.
- Setting the discriminant equal to zero for 'intersects' when tangency needs = 0 and intersection needs > 0.
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.
- Practice worksheet
- Practice worksheet — 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