IGCSE Additional Mathematics: Quadratic Functions Practice Questions
Completing the square gives the turning point and the range of a quadratic. The discriminant b squared minus 4ac decides the number of real roots: positive gives two, zero gives one repeated root, and negative gives none.
Topic 2 of Cambridge IGCSE Additional Mathematics 0606 is examined more heavily than almost any other, usually through the discriminant. Questions asking for a range of values of k are where the marks concentrate. The questions below build to those.
What you need to know for Quadratic Functions
- Completed square forma(x + b)2 + c. The turning point is at x equal to minus b, with y equal to c.
- Discriminantb2 minus 4ac. Greater than zero gives two distinct real roots, equal to zero gives one repeated root, less than zero gives no real roots.
- Equal rootsSet the discriminant equal to zero and solve for the unknown constant.
- No real rootsSet the discriminant less than zero, which produces an inequality to solve, often quadratic in the unknown.
- Maximum and minimumIf a is positive the parabola has a minimum, and if a is negative a maximum. The value is the constant in the completed square form.
IGCSE Additional Mathematics Quadratic Functions questions and answers
4 exam-style questions written to the 0606 syllabus. Try each one on paper first, then open the worked answer to check your method against the marks.
Express 2x2 minus 8x + 11 in the form a(x + b)2 + c.
Show the worked answer
- Take the coefficient of x2 out of the first two terms: 2(x2 minus 4x) + 11.
- Complete the square inside the bracket: x2 minus 4x = (x minus 2)2 minus 4.
- Substitute back: 2[(x minus 2)2 minus 4] + 11 = 2(x minus 2)2 minus 8 + 11.
- Simplify: 2(x minus 2)2 + 3, so a = 2, b = minus 2 and c = 3.
The equation x2 + kx + 9 = 0 has equal roots. Find the possible values of k.
Show the worked answer
- Equal roots means the discriminant is zero: b2 minus 4ac = 0.
- Here a = 1, b = k and c = 9, so k2 minus 4 x 1 x 9 = 0.
- k2 minus 36 = 0, so k2 = 36.
- k = 6 or k = minus 6. Both values give a perfect square trinomial.
Find the range of values of k for which x2 + kx + 9 = 0 has no real roots.
Show the worked answer
- No real roots means the discriminant is negative: b2 minus 4ac is less than 0.
- k2 minus 36 is less than 0, so k2 is less than 36.
- The critical values are k = 6 and k = minus 6, from the previous part.
- A quadratic in k opening upwards is below zero between its roots, so minus 6 is less than k and k is less than 6.
State the minimum value of 2x2 minus 8x + 11 and the value of x at which it occurs.
Show the worked answer
- From the completed square form, 2(x minus 2)2 + 3.
- The squared term is never negative, and its least value of zero occurs when x minus 2 = 0, so x = 2.
- At that point the function equals 3, so the minimum value is 3. It is a minimum rather than a maximum because the coefficient of x2 is positive.
Common mistakes in this topic
- Failing to multiply the completing the square correction by the factor outside the bracket.
- Giving only one value when a square root is taken.
- Choosing the wrong side of the critical values in a discriminant inequality.
- Confusing the conditions for equal roots and for no real roots.
- Reading the turning point without reversing the sign inside the bracket.
Exam tips
- Write down a, b and c with their signs before using the discriminant.
- Sketch the parabola in k when solving a discriminant inequality. It settles the interval instantly.
- Equal roots means discriminant equals zero. Two distinct roots means greater than zero. No real roots means less than zero.
- Completing the square gives the turning point, the range and the minimum in one step.
- Check a completed square form by expanding it back.
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Quadratic Functions FAQs
What does the discriminant tell you?
The discriminant, b squared minus 4ac, determines the number of real roots. Greater than zero gives two distinct real roots, equal to zero gives one repeated root, and less than zero gives no real roots. It is the standard route into any question about values of an unknown constant.
How do I complete the square when there is a coefficient in front of x squared?
Factorise that coefficient out of the first two terms only, complete the square inside the bracket, then multiply the correction term by the factor when you expand back out. Forgetting to multiply the correction is the most common error.
How do I find the range of values for no real roots?
Set the discriminant less than zero, which usually gives a quadratic inequality in the unknown. Find the critical values by solving the corresponding equation, then sketch the parabola to decide whether the solution lies between the roots or outside them.
How do I read the turning point from completed square form?
For a(x plus b) squared plus c, the turning point is at x equals minus b with y equals c. The sign inside the bracket reverses, so 2(x minus 2) squared plus 3 has its turning point at the point (2, 3).
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Written to the published Cambridge IGCSE Additional Mathematics (0606) syllabus. Check your school entry code and syllabus year, because Core and Extended candidates are assessed on different content. Last reviewed 2026-08-12.