Does a quadratic always have two roots?

Over the reals it may have one repeated root or none; the discriminant b² − 4ac decides.

Why do square roots take both signs?

Both 7 and −7 square to 49, so one square gives two possible roots.

S4 Quadratic Equations, Paper 1: factorising and the formula

2026-11-02

Try factorising first and fall back on the formula or completing the square; a quadratic usually has two roots, so square roots take both signs.

Focus　Rearrange every equation into ax² + bx + c = 0 before choosing a method.

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 − 5x + 6 = 0.
FoundationSolve x² = 49.
FoundationSolve x² + 3x = 0.
FoundationSolve (x − 4)² = 9.
IntermediateSolve 2x² − 7x + 3 = 0.
IntermediateSolve x² − 6x + 4 = 0 by completing the square, leaving surds.
IntermediateIf x² + kx + 9 = 0 has a repeated root, find k.
ChallengeIf α and β are the roots of 2x² − 5x + 1 = 0, find α + β, αβ and α² + β².
ChallengeForm a quadratic equation with integer coefficients whose roots are 3 and −1/2.
Error redoError: a student divides both sides of x² = 3x by x and gets x = 3. What went wrong?

03　Answers and one-line notes

#	Answer	One-line note
1	x = 2 or x = 3	Factorise as (x minus 2)(x minus 3) equals 0, so x is 2 or 3.
2	x = 7 or x = −7	Taking the square root of both sides gives plus or minus 7.
3	x = 0 or x = −3	Take out the common factor x to get x(x + 3) equals 0, so the roots are 0 and minus 3.
4	x = 7 or x = 1	Square roots give x minus 4 equals plus or minus 3, so x is 7 or 1.
5	x = 3 or x = 1/2	The discriminant is 49 minus 24, which is 25, so the formula gives 7 plus or minus 5 over 4.
6	x = 3 ± √5	Completing the square gives (x minus 3) squared equals 5, then take roots.
7	k = 6 or k = −6	A repeated root needs the discriminant k squared minus 36 to be zero, so k is plus or minus 6.
8	α + β = 5/2, αβ = 1/2, α² + β² = 21/4	Use sum equals minus b over a and product equals c over a, then (sum) squared minus twice the product.
9	2x² − 5x − 3 = 0	The sum is 5/2 and the product is minus 3/2, giving x squared minus 5x/2 minus 3/2 equals 0, then multiply by 2.
10	The root x = 0 is lost; x(x − 3) = 0 gives x = 0 or 3	Never divide by x, which may be zero; move all terms to one side 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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