Try factorising first (see how to factorise without guessing): it is the fastest method when it works. If the quadratic does not factorise, or the question says "give your answers correct to 2 decimal places", use the quadratic formula x = (−b ± √(b² − 4ac)) / 2a, which solves every quadratic. Use completing the square only when the question asks for it directly, or asks for a maximum or minimum value.
Students who know all three methods still lose marks, not because the algebra is wrong, but because they pick the wrong method for the question in front of them, or waste minutes trying factorising on a quadratic that was never going to factorise nicely. Each method solves the same equation. The exam question tells you which one it wants, if you read the wording carefully.
This guide is the decision layer that sits above the three individual methods. For the full factorising steps see how to factorise quadratic expressions without guessing, and for the full completing-the-square steps see completing the square. Here, the question is simply: which one do I reach for?
Try first
Factorising: fast, whole-number brackets, no wording clue needed
Default fallback
Quadratic formula: always works, use when factorising fails or 2 dp is asked for
Only when told
Completing the square: question says "complete the square" or asks for max/min
Watch the wording
"Correct to 2 decimal places" = signal to use the formula
In practice: scan for a factor pair for about 10 to 15 seconds. If nothing obvious appears, or the question already tells you the answers are not whole numbers, move straight to the quadratic formula. Only reach for completing the square when the question explicitly asks you to, or asks for the maximum or minimum value of an expression, since that is the one thing factorising and the formula cannot give you directly.
Solve x² − 2x − 15 = 0.
Solution
(x − 5)(x + 3) = 0x − 5 = 0 → x = 5x + 3 = 0 → x = −3Quick check: expand (x − 5)(x + 3) = x² + 3x − 5x − 15 = x² − 2x − 15. Matches the original, so the factor pair was correct. See the full split-the-middle-term method in this guide for cases where a is not 1.
Solve x² + 4x − 7 = 0, giving your answers correct to 2 decimal places.
The wording "correct to 2 decimal places" is the signal: this quadratic is not designed to factorise into whole numbers, so go straight to the quadratic formula rather than hunting for a factor pair that does not exist.
Solution
x = (−4 ± √(4² − 4(1)(−7))) / 2(1)4² − 4(1)(−7) = 16 + 28 = 44.x = (−4 ± √44) / 2. Since √44 ≈ 6.6332, this gives x = (−4 + 6.6332) / 2 or x = (−4 − 6.6332) / 2.x = 1.32 or x = −5.32 (2 decimal places).The step students get wrong
Substituting a negative b into −b is where most marks are lost. If b = 4, then −b = −4, not +4. If b were negative, for example b = −4, then −b becomes +4. Write out −b as its own line before substituting the rest of the formula, and never round the discriminant or any intermediate value: round only the final two answers, otherwise the 2 decimal place answer can be wrong in the second decimal.
Completing the square rewrites the quadratic as a(x + p)² + q. Neither factorising nor the formula gives you this form directly, which is why the question has to ask for it by name, or ask for the maximum or minimum value of a quadratic expression or curve. If you see either of those two signals, go to completing the square for the full worked steps rather than trying to force factorising or the formula to answer a "find the minimum value" question.
Bonus tip: on the fx-97SG (CW or X), Equation mode's Polynomial / Degree 2 option will solve any quadratic instantly and show you whether the roots are whole numbers, fractions, or decimals. Use it to decide which method to write up: whole-number roots mean factorising will work cleanly; decimal roots mean the formula is the write-up method the question expects.
How do I know a quadratic will not factorise, without wasting time trying?
Work out the discriminant b² − 4ac first. If it is a perfect square (0, 1, 4, 9, 16, 25, and so on), the quadratic factorises neatly into rational numbers (whole numbers when a = 1, as in most of the examples above). If it is positive but not a perfect square, like 44, it has two real roots but they will be decimals, so go straight to the quadratic formula.
Can I always use the quadratic formula, even when factorising would work?
Yes, the formula always works. Many students use it as a safety net when they are unsure whether a quadratic factorises. The tradeoff is time and arithmetic risk: factorising is usually faster and the answer is less likely to have a sign or rounding error when the numbers are simple.
What does "give your answer correct to 2 decimal places" actually tell me?
It tells you the answers are not exact whole numbers or simple fractions, so the examiner expects you to use the quadratic formula (or complete the square and simplify a surd) and round at the end. Seeing this phrase before you attempt factorising saves several minutes of guessing.
Is completing the square ever faster than the formula?
Not usually for solving an equation. Completing the square is essential, though, when the question asks for the maximum or minimum value of a quadratic, or the coordinates of the turning point, since that information is not visible from the formula's roots alone.
— Mr Gan Math Tuition
Mr. Gan teaches students to read the question wording and pick the right method the first time, instead of trying all three under exam pressure.
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