What does a zero discriminant mean?

The equation has one repeated real root and the graph just touches the x-axis.

What goes in the denominator of the formula?

It is 2a rather than 2; dropping the a is the commonest slip.

S4 Quadratic Equations, Paper 2: the formula and the roots

2026-11-02

Tutor Prep Guide · Primary English

The quadratic formula handles equations that will not factorise, while the discriminant says how many real roots exist.

Focus　Judge the roots with the discriminant, solve with the formula or by taking square roots, then check the sum and product.

01　Paper details (check answers afterwards)

Item	Detail
Questions	10
Suggested time	25
Difficulty mix	Foundation 4 · Intermediate 3 · Challenge 2 · Error redo 1
Chapter	Chapter: S4 Maths: Quadratic Equations
Level	S4

02　Questions (10 items)

FoundationSolve x² + 2x − 15 = 0 by factorising.
FoundationSolve 3x² = 12.
FoundationSolve x² − 6x = 0.
FoundationSolve (x + 3)² = 16.
IntermediateSolve x² − 4x − 1 = 0 and leave the answer in surd form.
IntermediateSolve 2x² + 5x − 3 = 0 using the formula.
IntermediateThe equation x² − 8x + k = 0 has equal roots. Find k.
ChallengeIf α and β are the roots of 2x² − 3x − 4 = 0, find α + β, αβ and α² + β².
ChallengeForm the quadratic equation whose roots are 1 + √2 and 1 − √2.
Error redoError: solving x² − 2x − 8 = 0, a student writes (x − 4)(x + 2) = 0 and answers x = 4 or x = 2. What is correct?

03　Answers and one-line notes

#	Answer	One-line note
1	x = −5 or x = 3	Factorising gives (x plus 5)(x minus 3), so the roots are minus 5 and 3.
2	x = 2 or x = −2	Divide both sides by 3 to get x squared equals 4, then take both square roots.
3	x = 0 or x = 6	Take out the common factor x to get x(x minus 6); dividing by x would lose the root zero.
4	x = 1 or x = −7	Taking the square root gives x plus 3 equals plus or minus 4, so 1 and minus 7.
5	x = 2 ± √5	The discriminant is 16 plus 4, giving 20, and the square root of 20 is 2 root 5.
6	x = 1/2 or x = −3	The discriminant is 25 plus 24, which is 49, and the denominator is 2a, namely 4.
7	k = 16	Equal roots mean the discriminant 64 minus 4k is zero, so k is 16.
8	3/2, −2 and 25/4	Roots and coefficients give a sum of 3 over 2 and a product of minus 2, then α² plus β² is 9/4 plus 4.
9	x² − 2x − 1 = 0	The sum is 2 and the product is minus 1, so the equation is x squared minus 2x minus 1.
10	x = 4 or x = −2	Setting x plus 2 to zero gives minus 2, so the sign has to flip.

04　What to do after marking

After marking, redo only the wrong items and write down which step went wrong (formula, unit, or misread number).
Repeat the same item two days later; move to the next paper only when both attempts are correct.
Parents only need the answer and note columns; there is no need to explain every item.

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