When to reject a root in A Math: logarithms, surds, exponentials and trigonometry

Many A Math equations are solved by turning them into a quadratic. The quadratic has two roots, but the original equation does not always accept both. Leaving an invalid root in the final answer usually costs the final answer mark, even when every line of method is right. The fix is one line of working: say which root fails, and why.

1. Logarithms: the argument must be positive

Check every root in the original equation, not in the combined one. Here x = −4 makes log₂ x undefined, even though it satisfies the quadratic.

2. Surds: squaring can add a root

Squaring both sides makes √(x + 7) = −(x − 5) look like a solution too. Substituting back is the only way to tell. A square root is never negative, so any root that makes the right-hand side negative must go.

3. Exponentials: a power of a positive number is positive

4. Trigonometry: the range, and dividing by zero

Two different traps. First, answers outside the stated range must be left out, and answers inside it must all be found. Second, dividing both sides by a trigonometric expression loses the solutions where that expression is zero.

Dividing by sin x instead would leave only cos x = 1, and lose x = 180°. Factorise; never divide by something that can be zero.

The habit that saves the mark

1. Before solving, note the conditions. Logs need positive arguments, square roots are never negative, powers of positive numbers are positive, and the question states a range.
2. After solving, test each root against them. Write the rejection as a line of working, with its reason.
3. In context questions, reject what cannot happen. A negative length or time is rejected for the same reason, and saying so is part of the answer.

These topics are in the SEAB O-Level Additional Mathematics 4049 syllabus (2026). For how method and answer marks are awarded, see How method marks work.