Sec 3 A-Math Quadratic Functions & Equations

Q: How does the discriminant tell me about a line and a curve?

Substitute the line into the curve to get a quadratic in x. Positive discriminant: two intersection points. Zero: the line is a tangent. Negative: no intersection. The same three cases as roots of a quadratic equation.

Q: When is this chapter usually tested in school?

Sec 3 schools typically assess this material in Weighted Assessment 1 (Term 1) and again in the end-of-year examination. Scope and weighting vary by school, so check your own school’s assessment outline.

Overview

What you’ll learn

Complete the square for ax² + bx + c and use the result to find the maximum or minimum value and where it occurs.
Decide when y = ax² + bx + c is always positive (or always negative) using the sign of a and the discriminant.
Set up and use quadratic functions as models, interpreting extreme values in context.
Use the discriminant b² − 4ac to determine whether a quadratic equation has two real roots, two equal roots or no real roots, and apply the same conditions to a line intersecting, touching or missing a curve.
Solve simultaneous equations in two unknowns by substitution when one equation is linear.
Solve quadratic inequalities and represent the solution set on a number line.

In 30 seconds

1Completing the square turns ax² + bx + c into a(x + h)² + k; the extreme value is k, at x = −h.
2Factor out the coefficient of x² before completing the square — the −4-style correction inside the bracket gets multiplied by a.
3a > 0 with a negative discriminant means the quadratic is always positive; a < 0 with a negative discriminant means always negative.
4Discriminant b² − 4ac: positive → two real roots, zero → two equal roots, negative → no real roots.
5Substituting a line into a curve gives a quadratic: the discriminant then tells you two points of intersection, tangency, or none.
6Solve linear–quadratic pairs by substitution: expect up to two solutions, and give the pair (x, y) for each.
7For ax² + bx + c ≤ 0 with a > 0 the solution set lies between the roots; for ≥ 0 it lies outside them (including the roots).
8Quadratic models: the extreme value usually answers the question — read the question again before stating it, with units.

Notes · 1 of 6

Completing the square and extreme values

Rewriting a quadratic as a(x + h)² + k exposes its largest or smallest value in one line.

A squared number is never negative, so a(x + h)² + k can dip no lower than k (when a > 0) and climb no higher than k (when a < 0). Completing the square rewrites any quadratic into this form.

1Factor out a from the x terms
ax² + bx + c = a(x² + (b/a)x) + c. This step is where most marks are lost when a ≠ 1.
2Complete the square inside
x² + px becomes (x + p/2)² − (p/2)². Keep the correction term inside the bracket.
3Multiply back out
Multiply the correction term by a before combining with c: a[(x + p/2)² − (p/2)²] + c = a(x + p/2)² + c − a(p/2)².
4Read off the extremes
Minimum (a > 0) or maximum (a < 0) value is the constant term; it occurs at x = −h, where h is half the original b/a.

The same form answers "state the coordinates of the minimum point" questions: the vertex sits at (−h, k). For f(x) = 2(x − 3)² − 13 the minimum point is (3, −13) — the x value is the one that makes the bracket zero.

Notes · 2 of 6

Always positive or always negative

A quadratic keeps one sign everywhere exactly when its curve never touches the x-axis.

The curve y = ax² + bx + c is a parabola. When a > 0 it opens upward; when a < 0 it opens downward. If it never meets the x-axis, every y value carries the sign of a.

Conditions on y = ax² + bx + c
Condition	Sign of a	Discriminant b² − 4ac
Always positive	a > 0	b² − 4ac < 0
Always negative	a < 0	b² − 4ac < 0
Never negative (zero allowed)	a > 0	b² − 4ac ≤ 0

Solving k² < 36 on a number line: k² − 36 = (k − 6)(k + 6) < 0 holds between the roots, so −6 < k < 6. Watch the wording: "always positive" excludes zero, "never negative" includes it — the strictness of the inequality is a mark.

Notes · 3 of 6

Using quadratic functions as models

Height, area and profit questions hide a parabola — the model converts the algebra into a practical answer.

A model is a formula linking two quantities. When the formula is quadratic, questions ask for extreme values, when they happen, and what they mean. Three steps cover almost every model question:

1Set up or verify the model
Define the variables, and show the formula is what the question claims (a "show that" part needs complete working).
2Complete the square
Convert to a(x − h)² + k to expose the extreme value k and the x value −h.
3Interpret in context
State the maximum height in metres, the best price in dollars, the largest area in cm² — with units, and check the value makes sense (areas and heights cannot be negative).

Model questions often add a restriction such as t ≥ 0. The algebra stays the same, but the interpretation must respect it — for example, a negative t answer is rejected as "not meaningful in context", and that sentence itself earns the interpretation mark.

Notes · 4 of 6

The discriminant: roots and line-curve conditions

One expression, b² − 4ac, decides whether roots exist, coincide or vanish — for equations and for line-curve pairs.

The quadratic formula x = (−b ± √(b² − 4ac)) / (2a) contains the discriminant b² − 4ac under the square root. Its sign decides everything:

What the discriminant tells you
b² − 4ac	Roots of the equation	Meaning on the graph
Positive	Two distinct real roots	Curve crosses the x-axis twice
Zero	Two equal real roots (a repeated root)	Curve touches the x-axis at one point
Negative	No real roots	Curve never meets the x-axis

The same test applies when a line meets a curve. Substitute the line into the curve to get a quadratic in x; the discriminant then reports two intersections, one (tangency) or none. A tangent line is exactly the discriminant = 0 case.

Notes · 5 of 6

Simultaneous equations: linear meets non-linear

Substitution turns a linear–quadratic pair into one quadratic equation in one unknown.

When one equation is linear, solve it for one variable and substitute into the other equation. The result is a quadratic, so expect up to two solution pairs (x, y).

1Rearrange the linear equation
Make one variable the subject, e.g. from y = x + 3 or x − 2y = 1.
2Substitute everywhere
Replace that variable in the quadratic equation — brackets around the substituted expression prevent sign slips.
3Solve the quadratic
Factorise, complete the square or use the formula. Keep exact values (fractions, surds) unless asked to round.
4Find both coordinates
Substitute each x value back into the linear equation (safer than the quadratic) to get the matching y values.

A "show that" variant asks you to derive a given quadratic, then solve it. Show every line: substitution, expansion, collection of terms, and the division by a common factor if you make one. Skipping the division line hides working and costs marks.

Notes · 6 of 6

Quadratic inequalities and number lines

Sketch the parabola in your head: the answer is between the roots or outside them.

To solve an inequality such as ax² + bx + c ≤ 0: find the roots of the matching equation, then use the sign of a to decide where the expression takes the wanted sign.

Choosing the region (a > 0, roots r₁ < r₂)
Inequality	Solution set	Number line
ax² + bx + c < 0	r₁ < x < r₂ (between the roots)	Open circles, segment shaded
ax² + bx + c ≤ 0	r₁ ≤ x ≤ r₂	Filled circles, segment shaded
ax² + bx + c > 0	x < r₁ or x > r₂ (outside the roots)	Open circles, outer rays shaded
ax² + bx + c ≥ 0	x ≤ r₁ or x ≥ r₂	Filled circles, outer rays shaded

When one root is a fraction or surd, the number line still uses the exact value — place the dot roughly at 0.5 for x = 1/2, at about 3.24 for x = 3 + √5. Integer questions ("state the largest integer in the solution set") must be answered by checking against the exact boundary, never by eyeballing the sketch.

Revise

Key terms

The words examiners expect you to use precisely.

Quadratic function: A function of the form y = ax² + bx + c with a ≠ 0; its graph is a parabola.
Completing the square: Rewriting ax² + bx + c as a(x + h)² + k to expose the extreme value and its location.
Discriminant: The expression b² − 4ac from the quadratic formula; its sign counts the real roots.Positive: two roots; zero: two equal roots; negative: none.
Two equal roots: The case b² − 4ac = 0, where the equation has one repeated solution, e.g. x² + 2x + 1 = 0 at x = −1.
Tangent: A line that touches a curve at exactly one point; algebraically, the substitution quadratic has discriminant zero.
Simultaneous equations: A pair of equations solved together; here one linear, one non-linear, solved by substitution.

Show 5 more key terms

Quadratic inequality: An inequality involving a quadratic expression, such as x² − 5x + 4 ≤ 0, with a range of solutions.
Solution set: All values of the variable that satisfy an inequality, shown as a range or on a number line.
Model: A formula linking two real quantities, used to predict behaviour such as maximum height or best price.
Vertex: The turning point of a parabola, at (−h, k) after completing the square; the maximum or minimum point.
Always positive: A function taking strictly positive values for every real x; for a quadratic this needs a > 0 and b² − 4ac < 0.

Revise

Common mistakes

Where students most often lose marks in this chapter — and what to do instead.

Mistake: Completing the square without factoring out a first, e.g. turning 2x² + 8x − 7 into 2(x + 4)² − 7.
Half of the coefficient inside the bracket is used, not half of the original coefficient, and the correction term must be multiplied by a.
Instead: Factor first: 2(x² + 4x) − 7 = 2[(x + 2)² − 4] − 7 = 2(x + 2)² − 15.
Mistake: Dropping the correction term when completing the square, e.g. x² + 6x = (x + 3)².
(x + 3)² = x² + 6x + 9, three more than the original expression.
Instead: x² + 6x = (x + 3)² − 9: subtract the square of half the coefficient.

Show 9 more mistakes

Mistake: Saying a quadratic is always positive just because a > 0.
An upward parabola with real roots dips onto or below the x-axis, taking negative and zero values.
Instead: Always positive needs a > 0 AND b² − 4ac < 0.
Mistake: Using b² − 4ac > 0 for "has real roots" (or = 0 for tangency questions about two intersections).
"Has real roots" includes the repeated-root case, and "two distinct intersections" excludes it.
Instead: Match the phrase: two distinct roots/intersections → > 0; repeated root/tangent → = 0; no real roots/no intersection → < 0.
Mistake: Losing one solution when solving simultaneous linear–quadratic pairs.
The substitution produces a quadratic, which usually has two roots; each needs its own y value.
Instead: Solve the quadratic fully and pair each x with the y from the linear equation: a line generally cuts a curve twice.
Mistake: Substituting back into the quadratic equation to find y.
Squaring errors propagate; the linear equation gives y directly and unambiguously.
Instead: Find y from the linear equation (y = x + 3 above), then verify in the quadratic if time allows.
Mistake: Writing the solution of x² − 5x + 4 ≤ 0 as x ≤ 1 or x ≤ 4.
The two factors must have opposite signs for the product to be negative, which happens between the roots, not on one side.
Instead: With a > 0: ≤ 0 gives between the roots, ≥ 0 gives outside them. Sketch the parabola mentally.
Mistake: Dividing both sides of x² = 3x by x to get x = 3.
x could be zero, and dividing by zero is illegal — the solution x = 0 is lost.
Instead: Bring everything to one side: x² − 3x = 0, so x(x − 3) = 0 and x = 0 or 3.
Mistake: Giving the x value when the question asks for the maximum value (or the reverse).
f(x) = 2(x − 3)² − 13 has minimum value −13, occurring at x = 3 — two different answers to two different questions.
Instead: Underline the question word: "value" → the constant k; "where/at what x" → −h; "coordinates" → the pair (−h, k).
Mistake: Rounding surd answers in "exact value" questions, e.g. writing 3 ± 2.24 instead of 3 ± √5.
A decimal is never exact unless the number terminates; the mark is for the surd form.
Instead: Keep surds and fractions exact until a question explicitly asks for 3 significant figures.
Mistake: Interpreting a model without units or context, e.g. answering "the maximum is 13.5" with no metres.
The final interpretation mark rewards the answer in context; a bare number is incomplete.
Instead: State "the greatest height is 13.5 m" — quantity, unit and quantity name.

Check yourself

Quick quiz

18 questions with instant explanations. About 5 minutes.

Question 1 of 3

Question 1: current question
Question 2: not answered
Question 3: not answered

Complete the square for a quadratic

Express x² + 6x + 5 in the form (x + a)² + b.

Practise

Topical questions

Exam-style practice in three tiers, from recall to data questions.

5Foundation
5Standard
5Challenging

15 questions worth 60 marks, each with a model answer, mark scheme and examiner tips.

Practise topical questions

Exam practice

Practice paper

A 30-mark paper set out like a school paper, with a separate mark scheme. Give yourself 40 min.

Question paper

Chapter 1 Practice Paper

30 marks · 40 min · 5 pages

5 multiple-choice questions and structured questions with answer lines. Chapter 1 (A1 Quadratic functions, A2 Equations and inequalities).

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Mark scheme

Chapter 1 Practice Paper — mark scheme

Answers · Marking points · 3 pages

Model answers with a mark for each creditworthy point — for marking at home or in class.

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Where this chapter is tested

Each school sets its own scope — check your assessment outline.

Questions

Frequently asked questions

What does Sec 3 A-Math Chapter 1 cover?

It covers syllabus sub-topics A1 and A2 of O-Level 4049 / SEC G3 K341: completing the square for maximum and minimum values, conditions for a quadratic to be always positive or always negative, quadratic models, discriminant conditions for roots and line-curve intersections, linear-quadratic simultaneous equations, and quadratic inequalities with number-line solutions.

How do I complete the square when the x² coefficient is not 1?

Factor the coefficient out of the x-terms first, complete the square inside the bracket, then multiply the correction term by the coefficient when you tidy up. For 2x² − 12x + 5: 2(x² − 6x) + 5 = 2[(x − 3)² − 9] + 5 = 2(x − 3)² − 13.

When is a quadratic always positive?

Exactly when it opens upward (a > 0) and has no real roots (b² − 4ac < 0). Both conditions are needed; if the curve touches the x-axis, zero is not a positive value, so the discriminant must be strictly negative.

How does the discriminant tell me about a line and a curve?