What does this module cover in N(A)-Level A-Maths 4051?

It covers three outcomes in the Algebra strand of syllabus 4051: expressing a quadratic in completed-square form to find the turning point, line of symmetry and graph; using the discriminant $b^2 - 4ac$ to decide the nature of the roots and find unknown constants; and solving quadratic inequalities. The three outcomes share the same underlying quadratic and reinforce each other.

What does completing the square give you?

Completing the square rewrites $ax^2 + bx + c$ as $a(x + p)^2 + q$. From this form you read the turning point directly as $(-p, q)$ and the line of symmetry as $x = -p$. It also tells you whether the parabola has a minimum (when $a > 0$) or a maximum (when $a < 0$).

What does the discriminant tell you?

The discriminant is $b^2 - 4ac$. If it is positive there are two distinct real roots; if it is zero there is one repeated (equal) real root; and if it is negative there are no real roots. This decides the nature of the roots without solving the equation, and lets you find unknown constants from a root condition.

How do you solve a quadratic inequality?

Rearrange so one side is zero, factorise to find the critical values (the roots), then use a sketch of the parabola or a number line to read off the range. For an expression less than zero you want where the parabola is below the axis; for greater than zero, where it is above. The answer is a single range or two separate ranges.

How is the Algebra strand assessed in syllabus 4051?

Both Paper 1 and Paper 2 (each 1 hour 45 minutes, 70 marks, weighted 50 percent) draw on the whole syllabus, and candidates answer all questions. Quadratics appear often, both directly and inside larger questions, so be fluent with completing the square, the discriminant and inequalities.

SingaporeAdditional Mathematics

Quadratic functions and equations in N(A)-Level Additional Mathematics (4051): completing the square to find the turning point and sketch the parabola, the discriminant and the nature of the roots, and solving quadratic inequalities

Q: What does the discriminant tell you?

The discriminant is $b^2 - 4ac$. If it is positive there are two distinct real roots; if it is zero there is one repeated (equal) real root; and if it is negative there are no real roots. This decides the nature of the roots without solving the equation, and lets you find unknown constants from a root condition.

Q: How do you solve a quadratic inequality?

Rearrange so one side is zero, factorise to find the critical values (the roots), then use a sketch of the parabola or a number line to read off the range. For an expression less than zero you want where the parabola is below the axis; for greater than zero, where it is above. The answer is a single range or two separate ranges.

Q: How is the Algebra strand assessed in syllabus 4051?

Both Paper 1 and Paper 2 (each 1 hour 45 minutes, 70 marks, weighted 50 percent) draw on the whole syllabus, and candidates answer all questions. Quadratics appear often, both directly and inside larger questions, so be fluent with completing the square, the discriminant and inequalities.

An N(A)-Level Additional Mathematics (4051) overview of quadratic functions and equations in the Algebra strand. How completing the square reveals the turning point and lets you sketch the parabola, how the discriminant decides the nature of the roots, and how to solve quadratic inequalities, with links to every dot point.

Generated by Claude Opus 4.88 min readSEAB-4051 AlgebraUpdated 2026-06-13

Reviewed by: AI editorial process; not yet individually human-reviewed

Jump to a section

One curve, three views
Completing the square and the graph
The discriminant and the nature of roots
Solving quadratic inequalities
How this module is examined
Check your knowledge

One curve, three views

Quadratic functions are a cornerstone of the Algebra strand of N(A)-Level Additional Mathematics (SEAB 4051). The same parabola is examined three ways: through completing the square to find its turning point and shape, through the discriminant to count its roots, and through inequalities to find where it is positive or negative. Seeing them as views of one curve makes the topic coherent.

This overview links three dot points: completing the square and quadratic graphs, the discriminant and nature of roots, and solving quadratic inequalities. Each has its own worked answers and practice.

Completing the square and the graph

The completing the square outcome rewrites $ax^2 + bx + c$ as $a(x + p)^2 + q$ . The turning point is then $(-p, q)$ and the line of symmetry is $x = -p$ . When $a > 0$ the parabola opens upward with a minimum; when $a < 0$ it opens downward with a maximum.

The discriminant and the nature of roots

The discriminant and nature of roots outcome uses $b^2 - 4ac$ to classify the roots of $ax^2 + bx + c = 0$ . If $b^2 - 4ac > 0$ there are two distinct real roots; if $b^2 - 4ac = 0$ there is one repeated root; if $b^2 - 4ac < 0$ there are no real roots. A given root condition turns into an equation or inequality in the unknown constant.

Solving quadratic inequalities

The solving quadratic inequalities outcome finds the range of $x$ that satisfies an inequality. Rearrange to make one side zero, factorise to find the critical values, then sketch the parabola or use a number line to decide which range satisfies the inequality.

How this module is examined

Both papers, all questions. Paper 1 and Paper 2 (each 1 hour 45 minutes, 70 marks, 50 percent) cover the full syllabus, and you answer every question.
State the discriminant condition. Write $b^2 - 4ac$ with the correct sign condition before substituting; it shows the examiner your reasoning.
Use a sketch for inequalities. A quick parabola sketch with the roots marked prevents the common error of giving the complementary range.

Check your knowledge

Short questions across the three outcomes. Work them with full method, then check the solutions.

Express $y = x^2 + 4x + 1$ in completed-square form. (2 marks)
State the turning point of $y = (x - 2)^2 + 3$ . (1 mark)
Find the discriminant of $x^2 - 4x + 4 = 0$ and state the nature of its roots. (2 marks)
Solve $x^2 - 5x + 6 > 0$ . (3 marks)

Solutions

The four questions cover completing the square, reading the turning point, computing the discriminant, and solving a quadratic inequality.

Step 1: Express $y = x^2 + 4x + 1$ in completed-square form

To complete the square, halve the coefficient of $x$ to get $2$ , square it to get $4$ , then add and subtract that value inside the expression:

y = (x + 2)^2 - 4 + 1 = (x + 2)^2 - 3.

Final answer (Q1): $y = (x + 2)^2 - 3$ .

Step 2: Read the turning point from $y = (x - 2)^2 + 3$

In completed-square form $a(x + p)^2 + q$ , the turning point is at $(-p, q)$ . Here $p = -2$ and $q = 3$ , so the turning point is read off directly without any calculation:

\text{Turning point} = (2, 3).

Final answer (Q2): The turning point is $(2, 3)$ .

Step 3: Find the discriminant of $x^2 - 4x + 4 = 0$

Identify the coefficients $a = 1$ , $b = -4$ , $c = 4$ , then substitute into $b^2 - 4ac$ :

b^2 - 4ac = (-4)^2 - 4(1)(4) = 16 - 16 = 0.

A discriminant of zero means there is exactly one repeated (equal) real root.

Final answer (Q3): Discriminant $= 0$ ; one repeated real root.

Step 4: Solve $x^2 - 5x + 6 > 0$

Factorise the left-hand side to find the critical values, then decide which regions of the $x$ -axis satisfy the inequality. The parabola opens upward (positive leading coefficient), so it is above the axis outside the roots:

(x - 2)(x - 3) > 0, \quad \text{critical values } x = 2 \text{ and } x = 3.

The upward parabola is above the $x$ -axis for $x < 2$ and for $x > 3$ .

Final answer (Q4): $x < 2$ or $x > 3$ .

Sources & how we know this

Singapore-Cambridge GCE N(A)-Level Additional Mathematics (Syllabus 4051) — Singapore Examinations and Assessment Board (2026)