Quadratic functions in Additional Math 0606: common mistakes and examiner advice

Quadratic questions in Additional Math cover completing the square, the discriminant, and quadratic inequalities. The reports show that students know the methods. The marks are lost in the final step: the conclusion, the region, or the form of the answer. This guide collects what Cambridge examiners say students get wrong, based on recent reports.

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

Mistakes that cost marks

• The wrong region for an inequality. The most common error: finding the right critical values and then choosing the wrong region. Common wrong answers were 1 < x < 4 written as 1 > x > 4, "x > 1 and x > 4", or only x > 4, and some students assumed the solution lay between the critical values when it did not. Others offered only the critical values.
• Final answer form. Writing a final answer such as 1 < x < 2.5 when the work led to something different, and losing the last mark. Reports also say two separate inequalities joined by "and" earn full credit.
• Using a sketch. The reports say candidates who used a sketch graph to confirm the solution set were more commonly correct, and that incorrect inequalities for k often came from not drawing a sketch.
• The discriminant. Not using the result in the concluding statement (for example, showing that 4k² + 33 = 0 has no real solution, and saying why), writing something such as "you cannot have a negative square root" without linking it to the discriminant, or being unable to recall b² − 4ac, which is in the list of formulas on page 2. Some used the wrong condition for two distinct roots.
• Setting up the equation. Setting the quadratic expression f(x) < 0 instead of solving f(x) = 0 first, or sign errors with the coefficient. When the question gives two curves or a line and a curve, the simplest method is to equate them, form a quadratic equal to zero and use the discriminant.
• Completing the square. Errors in a, b or c, such as a denominator of 25 on c, an incorrect expression in the brackets (for example 5(x² − 7/5)), omitting the power 2, or equating the expression to 0. In another question students used calculus to find the turning point when completing the square was asked for.
• "Hence". The word means use the result of the previous part. In one question students restarted with differentiation.
• Equations with powers. Cubing each term separately, or squaring 2x and 12 separately, and a belief that taking logs is the starting point whenever indices appear.
• Slips. Sign and arithmetic errors with the first expression, and bracketing errors that led to the wrong value of r.
• When the equation is wrong. The reports say the method of solving it must be seen for further marks to be awarded.

What examiners advise

• Draw a sketch of the quadratic to pick the correct region.
• Give the final answer as an inequality in the proper form, including the two parts for an "outside" region.
• Write the discriminant condition, and write a conclusion that links the result to the number of roots.
• If a question says "hence", use the previous result.
• Re-read the question before moving on, to check that every part has been answered.

Habits that help

• Before solving an inequality, sketch the curve and mark the critical values.
• Write b² − 4ac with the numbers inserted, then simplify one step at a time.
• Check a completed-square form by expanding it again.

Questions students ask

How do I choose the right region for a quadratic inequality?

Sketch the quadratic and mark the critical values. The most common error is finding the correct critical values and then choosing the wrong region. 1 < x < 4 written as 1 > x > 4, or "x > 1 and x > 4", loses the mark.

How should I write the final answer to an inequality?

As an inequality in the proper form, including the two parts for an "outside" region. Two separate inequalities joined by "and" can earn full credit where appropriate.

How do I use the discriminant in a conclusion?

Write the discriminant condition, then link the result to the number of roots. For example, show that 4k² + 33 = 0 has no real solution and say why, rather than "you cannot have a negative square root".

What goes wrong when completing the square?

Errors in a, b or c, a wrong expression in the brackets, omitting the power 2, or equating the expression to 0. When the question says "hence", use the result of the previous part.

Is the method seen if my equation is wrong?

The method of solving it must still be seen for further marks to be awarded.