Why is squaring in inequalities a common trap?

Because x^2 depends only on |x|, taking a root yields both signs; writing one direction gives only half the solution.

What makes an inequality true for all real x?

The curve must lie wholly above the axis: the leading coefficient is positive and the discriminant is negative, both at once.

How do tangency and crossing differ in the discriminant?

Tangency has a zero discriminant, one shared point; crossing has a positive discriminant, two shared points.

S4 Maths quadratic graphs paper 3: parameters, ranges and always-positive conditions

2026-11-10

Tutor Prep Guide · S4 Maths Quadratic Graphs

Ten items move to parameters: finding k from a repeated or two distinct roots, a range from a no-contact condition, the twin conditions for an always-positive inequality, and inequality endpoints.

Direct answers　Items 01 to 03 cover a repeated root for k, a range for two distinct roots and a greater-or-equal inequality, 04 to 06 finding a from one point, writing an equation from two roots and a line intersection, 07 to 09 a between-roots set, a no-contact range and an always-positive condition, and 10 a missing negative branch.

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)

FoundationThe equation x^2 + kx + 9 = 0 has a repeated root. Find the possible values of k.
FoundationThe equation x^2 + (k – 1)x + 4 = 0 has two distinct real roots. Find the range of k.
FoundationSolve the inequality x^2 – 4x + 3 >= 0.
IntermediateThe curve y = a(x – 2)^2 + 5 passes through (0, 1). Find a.
IntermediateA quadratic with leading coefficient 1 has roots 2 and -3. Write the equation.
IntermediateFind where y = x^2 – 4x + 1 meets y = x – 3.
ChallengeSolve the inequality (x – 1)(x + 3) < 0.
ChallengeThe curve y = x^2 – 4x + k does not meet the x-axis. Find the range of k.
ChallengeThe inequality kx^2 – 2x + 1 > 0 holds for every real x. Find the range of k.
Error redoError drill: solve x^2 >= 9. A student wrote x >= 3. What is the correct solution set?

03　Answers and one-line notes

#	Answer	One-line note
1	k = 6 or -6	A repeated root needs a zero discriminant: k^2 – 36 = 0, so k is 6 or -6, and both work.
2	k > 5 or k < -3	The discriminant (k – 1)^2 – 16 must be positive, so \|k – 1\| > 4 and k > 5 or k < -3.
3	x <= 1 or x >= 3	Factorising gives (x – 1)(x – 3) >= 0; the curve opens upward, so the expression is non-negative outside the roots, endpoints included.
4	a = -1	Substituting (0, 1): 1 = 4a + 5, so 4a = -4 and a = -1.
5	x^2 + x – 6 = 0	The sum is -1 and the product -6, so the equation is x^2 + x – 6 = 0, that is (x – 2)(x + 3) = 0.
6	(1, -2) and (4, 1)	Setting x^2 – 4x + 1 = x – 3 gives x^2 – 5x + 4 = 0, so x = 1 or 4, and the line gives the y values.
7	-3 < x < 1	The roots are 1 and -3 and the curve opens upward, so the product is negative between them.
8	k > 4	The discriminant 16 – 4k must be negative, so k > 4.
9	k > 1	The curve must open upward and stay above the axis: k > 0 together with discriminant 4 – 4k < 0 gives k > 1.
10	x >= 3 or x <= -3	Taking a square root gives \|x\| >= 3, so x >= 3 or x <= -3; writing only x >= 3 would wrongly drop values such as -4.

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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