O-Level Additional Mathematics: Quadratic Functions, Graphs & Discriminant-Based Problem Solving

Why Quadratic Functions Matter So Much in O-Level A-Math

Quadratic functions appear in almost every A-Math paper: graphs, inequalities, word problems, and even coordinate geometry. Students often know how to factorise but struggle when questions mix graphs, discriminant and context (e.g. area, time, height).

This guide focuses on three core skills:

• Reading key information from the quadratic equation
• Visualising and sketching the quadratic graph
• Using the discriminant to decide how many solutions (and what that means in context)

Core Forms of a Quadratic Function

At O-Level, you should be fluent in switching between these forms of a quadratic:

• General form: y = ax^2 + bx + c
• Factorised form: y = a(x - p)(x - q)
• Completed square (vertex) form: y = a(x - h)^2 + k

Each form gives different information quickly:

• General form: good for using discriminant b^2 - 4ac, and for comparing coefficients.
• Factorised form: roots / x-intercepts are visible: x = p and x = q.
• Completed square form: vertex is (h, k), and minimum/maximum value is easy to read.

Fast Conversion to Completed Square Form

Given y = ax^2 + bx + c (with a ≠ 0):

1. Factor out a from the first two terms: y = a[x^2 + (b/a)x] + c.
2. Inside the bracket, take half of the coefficient of x, square it, and add–subtract it: x^2 + (b/a)x = (x + b/2a)^2 - (b/2a)^2.
3. Simplify to get y = a(x + b/2a)^2 + (c - ab^2/4a^2).

In exams, you usually do this with numbers, not letters. Practise until you can complete the square in under 30 seconds.

Discriminant: Linking Algebra to the Graph

The discriminant Δ = b^2 - 4ac tells you how many real roots a quadratic equation ax^2 + bx + c = 0 has.

• Δ > 0: two distinct real roots (graph cuts x-axis at two points).
• Δ = 0: one repeated real root (graph touches x-axis at one point).
• Δ < 0: no real roots (graph does not meet x-axis).

In O-Level questions, this is often translated into phrases like:

• "No real solution" → Δ < 0
• "Exactly one solution" / "tangent" → Δ = 0
• "Two distinct solutions" / "intersects at two points" → Δ > 0

Example 1: Condition for a Tangent

Question: A line y = 2x + k is tangent to the curve y = x^2 + 4x + 1. Find k.

Method:

1. At intersection points, equate: x^2 + 4x + 1 = 2x + k.
2. Rearrange: x^2 + 2x + (1 - k) = 0.
3. For tangency, discriminant = 0: b^2 - 4ac = 0 with a = 1, b = 2, c = 1 - k.
4. Δ = 2^2 - 4(1)(1 - k) = 4 - 4 + 4k = 4k.
5. Set 4k = 0 → k = 0.

Exam point: Always rearrange to the form ax^2 + bx + c = 0 before applying the discriminant.

Vertex, Maximum/Minimum Value and Range

The sign of a tells you whether the quadratic has a maximum or minimum:

• a > 0: graph opens upwards → minimum point (vertex).
• a < 0: graph opens downwards → maximum point (vertex).

From completed square form y = a(x - h)^2 + k:

• Vertex is (h, k).
• If a > 0, minimum value of y is k, and y ≥ k.
• If a < 0, maximum value of y is k, and y ≤ k.

Example 2: Finding Range from Completed Square

Question: Express y = x^2 - 6x + 5 in the form a(x - h)^2 + k. Hence, state the range of y.

Solution:

1. Complete the square: y = x^2 - 6x + 5 = (x^2 - 6x + 9) - 9 + 5 = (x - 3)^2 - 4.
2. So y = (x - 3)^2 - 4. Here, a = 1 > 0, vertex is (3, -4).
3. Minimum value of y is -4 when x = 3.
4. Range: y ≥ -4.

Exam point: When the question says "Hence, state the range", they expect you to use the completed square form, not trial and error.

Quadratic Inequalities: Using the Graph Instead of Guessing

Many students try to solve inequalities like equations and then get confused. The systematic way:

1. Step 1: Solve the related equation ax^2 + bx + c = 0 to find the roots.
2. Step 2: Sketch a quick graph (shape and x-intercepts only).
3. Step 3: Use the graph to decide where the quadratic is positive/negative.

Example 3: Solving a Quadratic Inequality

Question: Solve x^2 - 5x + 4 < 0.

Solution:

1. Factorise: x^2 - 5x + 4 = (x - 1)(x - 4).
2. Roots are x = 1 and x = 4. Since a = 1 > 0, graph opens upwards.
3. Between the roots, the graph is below the x-axis (negative).
4. So x^2 - 5x + 4 < 0 when 1 < x < 4.

Exam point: For < 0, choose the region where the graph is below the x-axis. For > 0, choose where it is above.

Word Problems: Translating Context into a Quadratic

In O-Level A-Math, quadratic word problems often involve:

• Area of rectangles / fields
• Height vs time (projectile-like motion, but in A-Math context)
• Profit vs number of items

The process is always:

1. Define the variable clearly (e.g. x = length in m).
2. Form an expression (e.g. area, height, profit) in terms of x.
3. Recognise that expression as a quadratic function.
4. Use vertex / discriminant / roots depending on the question.

Example 4: Maximum Area Problem

Question: A rectangle has a perimeter of 40 m. Its length is x m and its breadth is y m.

1. Express the area A in terms of x.
2. Find the value of x for which the area is maximum, and state this maximum area.

Solution:

1. Perimeter: 2x + 2y = 40 → x + y = 20 → y = 20 - x.
2. Area: A = xy = x(20 - x) = -x^2 + 20x.
3. Write in completed square form: A = -(x^2 - 20x) = -(x^2 - 20x + 100) + 100 = -(x - 10)^2 + 100.
4. Here, a = -1 < 0, so vertex gives maximum area: maximum A = 100 when x = 10.
5. So maximum area is 100 m^2 when the rectangle is a square (10 m by 10 m).

Exam point: For "maximum" or "minimum" questions, always aim to express the quantity in completed square form.

Intersection of Two Quadratics or Line and Quadratic

Many questions ask for the number of intersection points between:

• a line and a quadratic, or
• two quadratic curves.

The steps are similar:

1. Equate the two expressions.
2. Rearrange to get a single quadratic equation.
3. Use discriminant to decide number of intersections.

Example 5: Line and Quadratic – Number of Intersections

Question: The curve y = x^2 - 3x + 2 and the line y = kx intersect at two distinct points. Find the range of values of k.

Solution:

1. At intersection: x^2 - 3x + 2 = kx.
2. Rearrange: x^2 - (3 + k)x + 2 = 0.
3. For two distinct points, discriminant > 0.
4. Δ = [-(3 + k)]^2 - 4(1)(2) = (3 + k)^2 - 8.
5. Condition: (3 + k)^2 - 8 > 0 → (3 + k)^2 > 8.
6. |3 + k| > √8 → 3 + k > √8 or 3 + k < -√8.
7. So k > √8 - 3 or k < -√8 - 3.

Exam point: Always interpret "intersect at two distinct points" as Δ > 0, "touches" as Δ = 0, and "no intersection" as Δ < 0.

Common Exam Mistakes with Quadratics

• Forgetting to rearrange to 0 before using the discriminant.
• Sign errors when completing the square or expanding.
• Mixing up maximum and minimum (not checking the sign of a).
• Writing range incorrectly (e.g. using > instead of ≥).
• Skipping the sketch for inequalities and relying on guesswork.

How to Practise Quadratic Functions Effectively

To be exam-ready for O-Level A-Math in Singapore, aim to:

• Drill 10–15 questions on completing the square until it is automatic.
• Practise mixed questions where you must decide whether to use discriminant, vertex or factorisation.
• Summarise on one page: general form, factorised form, completed square form, discriminant conditions, and typical word problem structures.

If you keep linking the algebra, graph and context together, quadratic questions become predictable and much easier to score full marks on.