This module treats a quadratic as one object with several faces: the standard form ax² + bx + c, the completed-square form, the factorised form and the graph. Additional Mathematics questions rarely ask you to “solve” only. They ask you to classify, to find a condition on a letter, to locate a turning point or to rebuild an equation from information.
Check the current Cambridge IGCSE Additional Mathematics 0606 syllabus for the exact wording and any given formulae in your exam year. The habits here stay the same across versions. Our Additional Mathematics learning guide shows where the module sits among the others.
What should you already know?
You need to expand brackets, factorise simple quadratics and solve them with the formula. You should be comfortable with fractions and negative numbers, because most errors in this module are sign errors. Functions from functions and restrictions help but are not required.
An orienting example
Consider y = x² − 6x + 5 and answer three things: its roots, its turning point and whether the line y = −5 touches it.
Roots: x² − 6x + 5 = (x − 1)(x − 5), so the roots are 1 and 5.
Turning point: complete the square. x² − 6x + 5 = (x − 3)² − 9 + 5 = (x − 3)² − 4. The turning point is (3, −4), and −4 is the minimum value.
The line y = −5: set x² − 6x + 5 = −5, which gives x² − 6x + 10 = 0. The discriminant is 36 − 40 = −4, which is negative, so the line never meets the curve. That agrees with the minimum of −4, since the curve never goes down to −5.
Check: at x = 3 the original gives 9 − 18 + 5 = −4, agreeing with the completed square.
One quadratic answered three different questions, and the three answers agree with each other. That consistency is the real skill of this module.
In which order should you study it?
- Complete the square without losing a coefficient: the form that exposes the turning point and minimum.
- Use the discriminant to classify intersections: tells you how many times a line meets a curve without solving anything.
- Recover a quadratic from roots: the reverse direction, using sum and product of roots.
- Connect a turning point with a minimum value: reads maximum and minimum values and their ranges from completed-square form.
- Solve a parameter condition for repeated roots: combines the discriminant with algebra in a letter.
Then try the mixed practice set. Move on to polynomial factors and remainders once repeated roots and factors feel natural.
Which traps catch students here?
- Forgetting to multiply the extra constant by the coefficient of x² when completing the square.
- Reading the discriminant from an equation that is not yet rearranged to ”= 0”.
- Writing (x + 3) as the root instead of x = −3, or flipping the sign in the sum and product of roots.
- Stating the minimum value as an x-coordinate, or the reverse.
- Treating “real and equal roots” as two different conditions.
How should you use the practice set?
Attempt each question on paper before opening the answer. Write the working as you would in an exam, then use the quadratic structure explorer to test your answer by changing coefficients and watching the graph.
Log every slip in the mistake log and retest queue by type (sign, coefficient, wrong form), so the next session targets the habit rather than the question. Students who want a teacher to read that log with them can look at online one-to-one Additional Mathematics tuition.