What does a repeated root look like on the graph?

The parabola just touches the x-axis at one point, so the discriminant is zero.

Why must one side be zero first?

Only when the other side is zero may each bracket be set to zero.

S4 Quadratic Equations, Paper 4: repeated roots, coefficients and consecutive numbers

2026-11-02

Tutor Prep Guide · Primary English

Paper 4 links factorising, the formula and the root relations, and the last item targets the non-zero right side trap.

Focus　Move everything to one side before factorising, test equal roots with the discriminant, and use 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 + 4x − 21 = 0.
FoundationSolve 3x² − 5x = 0.
FoundationSolve (x + 6)² = 25.
FoundationSolve x² − 8x + 16 = 0 and describe the roots.
IntermediateSolve x² + 3x − 5 = 0 with the formula, leaving surds.
IntermediateWithout solving, decide whether 2x² + 3x + 4 = 0 has real roots.
IntermediateIf x² + 8x + k = 0 has equal roots, find k.
Challengeα and β are roots of 2x² − 7x + 3 = 0. Find α + β, αβ and α² + β².
ChallengeThe product of two consecutive positive integers is 56. Find them.
Error redoError: solving (x − 2)(x + 3) = 6, a student writes x = 2 or x = −3. What is correct?

03　Answers and one-line notes

#	Answer	One-line note
1	x = 3 or x = −7	Factorising gives (x plus 7)(x minus 3), so the roots are minus 7 and 3.
2	x = 0 or x = 5/3	Taking out x gives x(3x minus 5), so one root is zero and the other is 5 over 3.
3	x = −1 or x = −11	The square root gives x plus 6 equals plus or minus 5, so minus 1 or minus 11.
4	x = 4, a repeated root	The left side is the perfect square (x minus 4) squared and the discriminant is zero, so 4 is a repeated root.
5	x = (−3 ± √29)/2	The discriminant is 9 plus 20, which is 29, so the formula gives minus 3 plus or minus root 29, all over 2.
6	No real roots, discriminant −23	The discriminant is 9 minus 32, which is minus 23, so the curve never meets the x-axis.
7	k = 16	Equal roots need the discriminant 64 minus 4k to vanish, so k is 16.
8	7/2, 3/2 and 37/4	The sum is 7 over 2 and the product 3 over 2, so the squares add to 49 over 4 minus 3, that is 37 over 4.
9	7 and 8	Let the smaller number be n, so n(n plus 1) is 56, giving (n plus 8)(n minus 7) and n equal to 7.
10	x = 3 or x = −4	Brackets may only be zeroed when the other side is zero; expand to x² plus x minus 12 and factorise instead.

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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