How do you write the solution set of a quadratic inequality?

Find the roots and note the opening direction: for an upward curve the expression is negative between the roots and positive outside; a downward curve reverses this.

What are the root-coefficient relations?

For ax^2 + bx + c = 0 the sum of the roots is -b / a and the product is c / a, usable without solving.

Can three points determine a parabola?

Yes; write f(x) = ax^2 + bx + c, substitute the three points and solve for a, b and c.

S4 Maths quadratic graphs paper 2: inequalities, root relations and three points

2026-11-10

Tutor Prep Guide · S4 Maths Quadratic Graphs

Ten items move to solution sets of quadratic inequalities, the root-coefficient relations, building a function from three points, and counting intersections with a line.

Direct answers　Items 01 to 03 cover solving, x-intercepts and a less-than inequality, 04 to 06 a greater-than inequality, a repeated root and three-point fitting, 07 to 09 intersection count, sum and product of roots and a completed-square minimum, and 10 the one-sided square root.

01　Paper details (check answers afterwards)

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

02　Questions (10 items)

FoundationSolve x^2 – x – 6 = 0.
FoundationFind the x-intercepts of y = 2x^2 – 8.
FoundationSolve the inequality x^2 – 5x + 6 < 0.
IntermediateSolve the inequality x^2 – 5x + 6 > 0.
IntermediateHow many x-axis contacts does y = x^2 + 6x + 9 have, and where?
IntermediateA quadratic graph passes through (0, 3), (1, 6) and (-1, 8). Find f(x).
ChallengeHow many points do y = x^2 and y = 2x + 3 share?
ChallengeGiven x^2 + 2x – 8 = 0, state the sum and the product of the roots.
ChallengeFind the minimum value of y = 3x^2 – 12x + 7.
Error redoError drill: solve x^2 < 4. A student wrote x < 2. What is the correct solution set?

03　Answers and one-line notes

#	Answer	One-line note
1	x = 3 or -2	Factorising gives (x – 3)(x + 2) = 0, so x = 3 or -2.
2	(2, 0) and (-2, 0)	Setting y = 0 gives x^2 = 4, so x = 2 or -2, placed symmetrically about the y-axis.
3	2 < x < 3	Factorising gives (x – 2)(x – 3) < 0; the curve opens upward, so it is negative between the roots.
4	x < 2 or x > 3	With the same roots and an upward curve, the expression is positive outside the roots.
5	One, at (-3, 0)	The discriminant is 36 – 36 = 0, a repeated root; factorising gives (x + 3)^2, so the contact is (-3, 0).
6	f(x) = 4x^2 – x + 3	The constant term is 3; substituting the two points gives a + b = 3 and a – b = 5, so a = 4 and b = -1.
7	Two (x = 3 and -1)	Setting x^2 = 2x + 3 gives x^2 – 2x – 3 = 0 with discriminant 4 + 12 = 16, so there are two points.
8	Sum = -2, product = -8	By the root-coefficient relations the sum is -b / a = -2 and the product is c / a = -8.
9	-5	Completing the square gives 3(x – 2)^2 – 5, and the squared term is least at zero, so the minimum is -5.
10	-2 < x < 2	Taking a square root gives both signs: x^2 < 4 means -2 < x < 2, while x < 2 would wrongly include values such as -5.

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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