Quadratics in A Level Math Paper 1: common mistakes and examiner advice

Quadratics sit underneath a large part of Paper 1. They appear as equations, inequalities, completed squares, discriminants, tangents, and as disguised equations inside trigonometry, indices and functions. This guide pulls together what Cambridge examiners say students get wrong, based on the reports from recent sessions.

Summarised in our own words from a reading of the examiner reports.

Mistakes that cost marks

• Not showing how the equation was solved. This is the most repeated comment on quadratics. Writing the roots straight from the calculator does not earn the method mark. Show the factorisation, the formula with the numbers substituted, or the completed square.
• Brackets that do not expand to the original. A factorisation whose brackets do not give back the coefficients of the quadratic is not accepted.
• Solving a disguised quadratic and stopping early. A substitution such as u = x² or u = x³ turns the equation into a quadratic. Students then solve for u and forget to go back: they do not take the square root (both positive and negative), they take a cube root when they should cube, or they do not work out the corresponding values of x at all.
• Dropping or mishandling a solution. Negative roots are lost. An answer such as x = 0 is lost by dividing through by x. A solution that is not valid for the question, such as a zero number of terms or a negative length, is left in without comment.
• Slips in forming the quadratic. Reports describe sign errors during rearrangement, putting both expressions equal to zero and then equating them, and squaring each term of a bracket instead of expanding it, so that (x + 1)² becomes x² + 1.
• Discriminant errors. The discriminant is usually known, but students make errors in the numbers (a bracket not squared, a sign lost when simplifying), forget the case where the discriminant equals zero, or use b² − 4ac with the wrong values of a, b and c.
• Inequalities. Answers are left as two numbers (such as −7 and 1) with no inequality, written as k > ±2, or given as the outside regions when the inside region is wanted. The inequality can also be strict when it should not be (or the reverse).
• Completing the square. The form is mixed up with a(x − p)² + q, the whole function is divided by the leading coefficient, the sign of the constant outside the bracket goes wrong, or a ± is left in the final answer. When asked for the least value, students often give the x value or the coordinates instead.

What examiners advise

• Use "hence" and the links between parts. Several reports ask students to think about why an earlier part was set. Completing the square in one part is often meant to help with the range, the turning point or the inverse in the next.
• Think about which form is quickest. The completed square gives the maximum or minimum, and the range, without calculus. Reports describe long solutions that could have been short.
• Check your answers in context. If a quadratic gives two answers, decide whether both make sense. A solution such as N = 0 in a context question should be questioned.
• Write the answer in the form requested. Use an inequality, not a list of numbers. Use y, or f(x), when describing a range, not x. Say which variable you found.
• Do not use a rounded decimal as an exact solution. If an exact value is required, use the formula or the completed square and leave surds in your answer.
• Check by substituting. For simultaneous equations, the question needs values of both x and y.

A short checklist before you leave a quadratics question

1. Have I shown the method for solving?
2. Do I have every solution, including negative roots and zero?
3. Did I return from a substitution to the original variable?
4. Is the final answer an inequality or a pair of values, as the question asks?
5. Have I checked each solution in the context?

Questions students ask

Why do I lose marks when I solve a quadratic with my calculator?

Examiners say this is the most repeated comment on quadratics. Writing the roots straight from the calculator does not earn the method mark. Show the factorisation, the formula with the numbers substituted, or the completed square, and check that your brackets expand back to the original quadratic.

I used a substitution such as u = x² and found u. Am I finished?

No. Go back to the original variable. Students lose marks by not taking both the positive and negative square root, by taking a cube root when they should cube, or by never working out the values of x at all.

How should I write the answer to a quadratic inequality?

As an inequality, not as a list of two numbers. Examiners report answers left as two values with no inequality, answers written as k > ±2, and the outside regions given when the inside region was wanted. Say which region the question needs.

If a question asks for the least value of a quadratic, what do I give?

The value of the function, not the value of x or the coordinates. When you complete the square, the number outside the bracket is the least value, and it occurs at the x value that makes the bracket zero.

Why do examiners say to check my quadratic answers in context?

A quadratic gives two answers, and one may not make sense, such as a zero number of terms or a negative length. Decide whether each solution fits the question, and say so if one is rejected.