Rearrange to , then pick a method
Every quadratic must first be written as . Then factorise if it factorises cleanly, use the quadratic formula for decimal answers, or complete the square for an exact (surd) answer or the turning point.
Factors are not the same as solutions
Factorising gives ; solving means setting each bracket to , so or . When a question says "hence solve", read the roots straight off the factors you already found — do not start again.
The discriminant counts the roots
If there are two real roots; if one repeated root; if no real roots. "Show it has no real solutions" is a discriminant argument, not a full solve.
Drawn from real examiner reports.
Completing the square when
When , the number produced inside the bracket must be multiplied by the leading coefficient. Students wrote , leaving when it should be , i.e. . For a negative leading coefficient, take out the (or ) first.
A commonly reported examiner note records the $2(x-6)^2-36+7$ slip, and (separately) not taking out the $-3$ first.
Re-solving after a 'factorise' part
After factorising in part (i), students asked to "hence solve" try to factorise again or use the quadratic formula instead of reading the roots from their factors — and some give factors when solutions were asked for (or vice versa).
A commonly reported examiner note records using the formula in the factorise part scoring no marks.
Signs the wrong way round in the brackets
Choosing the correct factor pair but placing the signs wrongly, e.g. writing for instead of . Expand back to check: , which is not . Multiplying the brackets out again instantly catches a sign slip.
A commonly reported examiner note records the main incorrect answer being the signs the wrong way round in the brackets; examiners advised multiplying back as a check.
Formula: dividing by 2 not
In the whole numerator is divided by , not by . Watch the sign of when is negative, keep the (two roots), and rearrange properly to first.
A commonly reported examiner note records students writing $2$ as the denominator instead of $2a$, and treating a constant on the right as $0$ without rearranging.
Not rearranging to = 0 first
A quadratic must read before you factorise or use the formula. Leaving a constant on the right — solving as though the right side were — gives wrong roots. Move every term to one side first.
Check factors by expanding back
Multiply your factors back out as a quick self-check — it catches a sign in the wrong bracket at once, e.g. should expand to the original .
Match your method to the answer form
Let the wording pick the method: "correct to 2 d.p." means the quadratic formula, an exact or surd answer means simplify , and "in the form " or a turning point means complete the square.
Reject roots that break the context
In a length, time or area context, or a stated domain such as , discard the root that cannot apply rather than quoting both — and state the reason. The rejection itself earns the final mark.
A quadratic equation must be written as before you solve it. A common slip is to leave a constant on the right (e.g. solving as if the right-hand side were ) — move everything to one side first.
What must you do before factorising or using the formula?
Solve .
