Use the discriminant to classify roots and intersections
Use b² − 4ac to decide whether a quadratic has two, one or no real roots and whether a line meets its graph.
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What is the discriminant of ax² + bx + c?
What you will learn
- Relate the discriminant to real roots
- Classify a line's intersections with a quadratic curve
- Connect the discriminant to the quadratic's minimum or maximum
The discriminant counts real roots
For ax² + bx + c = 0, the discriminant is Δ = b² − 4ac. If Δ > 0 there are two distinct real roots; if Δ = 0 there is one repeated real root; if Δ < 0 there are no real roots.
A line y = k intersects y = q(x) where q(x) − k = 0. Apply the same discriminant test to that equation. Two roots mean two intersections, a repeated root means tangency, and no real roots mean no intersection.
A turning point gives a quick check
For q(x) = x² − 4x + 1, complete the square to get (x − 2)² − 3. The minimum is −3, so horizontal lines below −3 miss the graph; y = −3 touches it; lines above −3 meet it twice.
This graph interpretation should agree with the discriminant for q(x) − k = 0.
Pause and explain
A line y = k is tangent to a quadratic at one point. What is the discriminant of q(x) − k = 0?
Worked example
Classify how y = x² − 4x + 1 meets y = k as k varies.
Show the worked solution
- Set x² − 4x + 1 = k, or x² − 4x + (1 − k) = 0.
- The discriminant is Δ = 16 − 4(1 − k) = 12 + 4k.
- Compare Δ with zero, or use the minimum value −3.
Answer Two intersections for k > −3; one tangent intersection for k = −3; no real intersection for k < −3.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
How many real roots does 2x² + 3x + 5 = 0 have?
Give me a hint
Calculate Δ before using the quadratic formula.
Compare my reasoning
- Δ = 3² − 4(2)(5).
- Δ = 9 − 40 = −31.
- A negative discriminant gives no real roots.
No real roots.
Look for these in your work
- Correct expression
- Arithmetic checked
- Conclusion matches sign
Find the values of m for which x² + 2x + m = 0 has equal real roots.
Give me a hint
Set the discriminant to zero.
Compare my reasoning
- Δ = 2² − 4(1)(m) = 4 − 4m.
- For equal roots, 4 − 4m = 0.
- Solve for m.
m = 1.
Look for these in your work
- Repeated-root condition used
- Equation formed
- Parameter solved
A circular arch is modelled by y = 10 − x², with ground level y = 0. Use the discriminant to determine how many ground contact points the model has.
Give me a hint
Set the height equal to zero and classify the quadratic roots.
Compare my reasoning
- Solve 10 − x² = 0, or x² − 10 = 0.
- The discriminant is 0² − 4(1)(−10) = 40 > 0.
- There are two distinct real x-values where the arch meets the ground.
Two ground contact points, at x = ±√10.
Look for these in your work
- Intersection equation set up
- Discriminant positive
- Two points interpreted in context
Common mistakes
- Reversing the repeated-root condition
- Applying the discriminant to the original curve instead of q(x) − k
- Calling a repeated root two distinct intersections
What can you explain now?
What does Δ = 0 mean for a quadratic equation?
Compare with the explanation
One repeated real root
The quadratic touches the x-axis at one x-value.
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Make it stick
State all three discriminant cases and their graph meanings.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.