What is a quadratic equation and how is it represented in Pure Mathematics 1?

A quadratic equation is a second-degree polynomial equation in a single variable x, with a non-zero coefficient for x². It is generally represented as ax² + bx + c = 0, where a, b, and c are constants. In the Cambridge International A Levels Pure Mathematics 1 curriculum, understanding the structure and solutions of quadratic equations is fundamental.

How do you solve a quadratic equation using the quadratic formula?

To solve a quadratic equation ax² + bx + c = 0 using the quadratic formula, apply the formula x = (-b ± √(b² - 4ac)) / (2a). This formula provides the solutions for x by calculating the roots of the equation. The term under the square root, b² - 4ac, is known as the discriminant and determines the nature of the roots.

What is the significance of the discriminant in a quadratic equation?

In the context of quadratic equations, the discriminant (b² - 4ac) helps determine the nature of the roots. If the discriminant is positive, the equation has two distinct real roots. If it is zero, there is exactly one real root (a repeated root). If the discriminant is negative, the equation has two complex roots. Understanding this is crucial for students studying Pure Mathematics 1 at the Cambridge International A Levels.

How can you factorize a quadratic expression?

To factorize a quadratic expression of the form ax² + bx + c, you need to express it as a product of two binomials. This involves finding two numbers that multiply to ac and add to b. Once these numbers are identified, the quadratic can be rewritten and grouped to facilitate factorization. This method is often taught in the Cambridge International A Levels Pure Mathematics 1 course.

What are the applications of quadratic equations in real-world scenarios?

Quadratic equations are widely used in various real-world applications, such as calculating projectile motion, optimizing areas and volumes, and modeling economic behaviors. In the Cambridge International A Levels Pure Mathematics 1 curriculum, understanding these applications helps students appreciate the practical relevance of quadratics in solving real-life problems.

Why is "Using $\geq$ instead of $>$ for distinct real roots" a common mistake in Quadratics?

Why it happens: Confusion with 'real roots' (which allows equality). How to avoid it: Two DISTINCT roots: > 0. Two real roots (possibly equal): ≥ 0. Equal roots: = 0. Source: 9709 Examiner Reports 2022-2024

Why is "Forgetting to factor coefficient from BOTH x² and x terms" a common mistake in Quadratics?

Why it happens: Rushing to complete the square. How to avoid it: For 2x^2 - 12x + 7, factor 2 from FIRST TWO terms only: 2(x^2 - 6x) + 7. The constant 7 is untouched. Source: 9709 Examiner Reports 2022-2024

Why is "Wrong direction for quadratic inequality solution" a common mistake in Quadratics?

Why it happens: Forgetting parabola shape. How to avoid it: For ax^2 + bx + c ≥ 0 with a > 0: solution is OUTSIDE the roots (x ≤ r_1 or x ≥ r_2). For ≤ 0: solution is BETWEEN the roots. Source: 9709 Examiner Reports 2022-2024

Pure Mathematics 1Cambridge International A Levels Mathematics (9709)

Quadratics

Work through the notes, try the practice questions, then take the quiz. The report tells you exactly what to revise next.

PreviousNext

Learn this Topic

Start with the slides for the quick version, then go deeper with the full study notes.

Short Study Notes in the form of Slides

Read the notes first. If the method in a worked example clicks, you're ready for the questions.

Page 1 / 0

Detailed Study Notes

Full prose, callouts and a recap — built for A* mastery, not just a quick scan.

Take these study notes with you

Download a branded PDF — full prose, callouts, recap and memorise list for Quadratics, ready to print or save offline.

Quadratics Study Notes — Cambridge International A Level Mathematics 9709 P1 (2024-2026 syllabus)

Completing the square. The discriminant. Quadratic inequalities. Simultaneous equations one quadratic. Everything you need to score full marks on P1 Q1.

What you’ll learn

Mapped to the Cambridge IGCSE 9709 syllabus (2024-2026).

P1.1.1 — Complete the square and use to find vertex.
P1.1.2 — Use discriminant to classify roots.
P1.1.3 — Solve quadratic equations and inequalities.
P1.1.4 — Solve simultaneous equations with one quadratic.

Completing the square

Rewrite as $a(x+h)^2 + k$ .

Method. Convert $ax^2 + bx + c$ to $a(x+h)^2 + k$ .

Steps.

Factor $a$ from the first two terms: $a(x^2 + \frac{b}{a}x) + c$ .
Half the coefficient of $x$ , square it: $h = \frac{b}{2a}$ , then $h^2$ .
Add and subtract $h^2$ inside the bracket: $a[(x + h)^2 - h^2] + c$ .
Simplify: $a(x+h)^2 - ah^2 + c$ .

Worked example. $2x^2 - 12x + 7$ .

Factor 2 from first two: $2(x^2 - 6x) + 7$ .
Half of $-6$ is $-3$ ; squared is $9$ : $2[(x-3)^2 - 9] + 7$ .
Expand: $2(x-3)^2 - 18 + 7 = 2(x-3)^2 - 11$ .

Why this is useful.

Vertex. Minimum (if $a > 0$ ) at $x = -h$ , $y = k$ . So $2(x-3)^2 - 11$ has min $-11$ at $x = 3$ .
Range. $y \geq -11$ for $a > 0$ ; $y \leq k$ for $a < 0$ .
Sketching. Tells you vertex and shape.

Cambridge tip. Mark schemes often reward "completed square form" PLUS extracting vertex and minimum from it.

Factor $a$ from first two terms.
Half-square-the-x-coefficient.
Vertex at $x = -h$ , $y = k$ .

See the full worked example for quadratics →

The discriminant

$\Delta = b^2 - 4ac$ .

$\Delta = b^2 - 4ac$ classifies the roots of $ax^2 + bx + c = 0$ :

$\Delta > 0$ : two distinct real roots (curve crosses x-axis twice).
$\Delta = 0$ : one repeated real root (curve touches x-axis).
$\Delta < 0$ : no real roots (curve does not cross x-axis).

Common use. Find values of parameter $k$ for given root condition.

Example. For $x^2 + (k+2)x + 9 = 0$ to have TWO DISTINCT real roots: $\Delta = (k+2)^2 - 36 > 0$ $(k+2)^2 > 36$ $k+2 > 6$ or $k+2 < -6$ $k > 4$ or $k < -8$ .

Going deeper. Discriminant of $ax^2 + bx + c$ relates to vertex height: minimum $y$ value (when $a > 0$ ) is $c - \frac{b^2}{4a} = -\frac{\Delta}{4a}$ . So $\Delta < 0$ ⟺ minimum > 0 (curve sits above x-axis).

Cambridge tip. Get the inequality direction right — distinct roots demand STRICT inequality $\Delta > 0$ .

$\Delta > 0$ : two distinct real roots.
$\Delta = 0$ : repeated root.
$\Delta < 0$ : no real roots.

See the full worked example for quadratics →

Quadratic inequalities

Sketch + identify regions.

Method.

Factorise (or use quadratic formula) to find roots $r_1, r_2$ .
Sketch the parabola: opens up if $a > 0$ , down if $a < 0$ .
Identify where the curve is above/below x-axis.

Patterns (for $a > 0$ , roots $r_1 < r_2$ ):

$ax^2 + bx + c > 0$ : outside the roots — $x < r_1$ or $x > r_2$ .
$ax^2 + bx + c < 0$ : between the roots — $r_1 < x < r_2$ .

Example. $x^2 - x - 6 \geq 0$ factorises as $(x-3)(x+2) \geq 0$ . Parabola opens up, roots at $-2$ and $3$ . Curve $\geq 0$ OUTSIDE roots: $x \leq -2$ or $x \geq 3$ .

For $a < 0$ , flip the regions.

Cambridge tip. ALWAYS SKETCH the parabola — examiner mistakes go away once you can see the shape.

Factorise, sketch, read regions.
$> 0$ : outside roots (for $a > 0$ ).
$< 0$ : between roots (for $a > 0$ ).

See the full worked example for quadratics →

Simultaneous equations (one quadratic)

Substitute line into quadratic.

Linear + quadratic system. Solve by substitution: rearrange the linear equation to make $y$ (or $x$ ) the subject, substitute into the quadratic.

Example. $y = x + 2$ and $y^2 = 4x + 12$ .

Sub: $(x+2)^2 = 4x + 12$ .
Expand: $x^2 + 4x + 4 = 4x + 12$ .
Simplify: $x^2 = 8$ , so $x = \pm 2\sqrt{2}$ .
Find $y$ : $y = x + 2$ , so $y = 2 \pm 2\sqrt{2}$ .

Geometric interpretation. Solutions are POINTS where the line meets the curve. Discriminant tells you HOW MANY intersections.

Cambridge tip. Always state SOLUTIONS AS POINTS $(x, y)$ .

Substitute linear into quadratic.
Reduces to one-variable quadratic.
Solutions are POINTS.

See the full worked example for quadratics →

How it’s examined

Quadratics is heavily tested on P1 — usually appears in Q1-Q3 of 9709/12 and 9709/13. Most-tested: completed square (5 marks), discriminant (5 marks), simultaneous equations (6-8 marks).

Sources: Cambridge International A Level Mathematics 9709 syllabus (2024-2026); 9709 Examiner Reports 2022-2024; 9709/12 May/Jun 2024 question paper and mark scheme. Last reviewed 2026-05-11.

Master this Topic

Worked examples, formulae, definitions and the mistakes examiners flag — everything you need to push from a pass to an A*.

Take this whole topic with you

Download a branded revision sheet — worked examples, formulae, definitions and common mistakes for Quadratics, ready to print or save as PDF.

Step-by-step worked examples — Quadratics

Step-by-step solutions to past-paper-style questions on quadratics, written exactly the way a tutor would explain them at the board.

1Completing the square (5 marks)
Extended• Adapted from 9709/12 May/Jun 2024 Q1• completing the square
Question
Express $2x^2 - 12x + 7$ in the form $a(x+b)^2 + c$ , where $a$ , $b$ and $c$ are constants. State the minimum value of the expression and the value of $x$ at which it occurs. (5 marks)
Step-by-step solution
1. Step 1
  Factor out the coefficient of $x^2$ from the first two terms.
  $2x^2 - 12x + 7 = 2(x^2 - 6x) + 7$
2. Step 2
  Complete the square inside the bracket. Half of $-6$ is $-3$ ; square it to get $9$ .
  $x^2 - 6x = (x-3)^2 - 9$
3. Step 3
  Substitute back and expand.
  $2[(x-3)^2 - 9] + 7 = 2(x-3)^2 - 18 + 7 = 2(x-3)^2 - 11$
4. Step 4
  Identify $a$ , $b$ , $c$ . $a = 2$ , $b = -3$ , $c = -11$ .
5. Step 5
  Minimum value. Since $2(x-3)^2 \geq 0$ , minimum is $-11$ at $x = 3$ .
Answer
$2(x-3)^2 - 11$ ; minimum value $-11$ at $x = 3$ .
Examiner tip
Common error: factor 2 from all three terms — DON'T. Only factor it from the $x^2$ and $x$ terms.
2Using the discriminant (5 marks)
Extended• discriminant
Question
Find the values of $k$ for which the equation $x^2 + (k+2)x + 9 = 0$ has two distinct real roots. (5 marks)
Step-by-step solution
1. Step 1
  Two distinct real roots requires $\Delta > 0$ where $\Delta = b^2 - 4ac$ .
2. Step 2
  Identify coefficients. $a = 1$ , $b = (k+2)$ , $c = 9$ .
3. Step 3
  Apply $\Delta > 0$ .
  $(k+2)^2 - 4(1)(9) > 0$
4. Step 4
  Expand and solve.
  $(k+2)^2 > 36 \Rightarrow k+2 > 6 \text{ or } k+2 < -6$
5. Step 5
  Final answer. $k > 4$ or $k < -8$ .
Answer
$k > 4$ or $k < -8$ .
3Simultaneous equations with quadratic (6 marks)
Extended• Adapted from 9709/12 Oct/Nov 2024• simultaneous
Question
Solve the simultaneous equations $y = x + 2$ and $y^2 = 4x + 12$ . (6 marks)
Step-by-step solution
1. Step 1
  Substitute the linear equation into the quadratic.
  $(x+2)^2 = 4x + 12$
2. Step 2
  Expand and rearrange.
  $x^2 + 4x + 4 = 4x + 12 \Rightarrow x^2 = 8$
3. Step 3
  Solve for $x$ . $x = \pm 2\sqrt{2}$ .
4. Step 4
  Substitute back for $y$ . $y = x + 2$ , so $y = 2 + 2\sqrt{2}$ or $y = 2 - 2\sqrt{2}$ .
Answer
$(2\sqrt{2},\, 2 + 2\sqrt{2})$ and $(-2\sqrt{2},\, 2 - 2\sqrt{2})$ .
4Quadratic inequality (4 marks)
Extended• inequality
Question
Solve the inequality $x^2 - x - 6 \geq 0$ . (4 marks)
Step-by-step solution
1. Step 1
  Factorise.
  $x^2 - x - 6 = (x-3)(x+2)$
2. Step 2
  Find critical values. $x = 3$ or $x = -2$ .
3. Step 3
  Sign analysis. Curve $y = (x-3)(x+2)$ opens upwards. Above x-axis when $x \leq -2$ or $x \geq 3$ .
4. Step 4
  Final answer. $x \leq -2$ or $x \geq 3$ .
Answer
$x \leq -2$ or $x \geq 3$ .

Key Formulae — Quadratics

The formulae you need to memorise for quadratics on the Cambridge International A Level 9709 paper, with every variable defined in plain English and a note on when to use it.

Quadratic formula
$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
$a, b, c$
coefficients in $ax^2 + bx + c = 0$
When to use
When solving $ax^2 + bx + c = 0$ and factorising is not straightforward.
Discriminant
$\Delta = b^2 - 4ac$
$\Delta$
discriminant
When to use
Determines nature of roots. $\Delta > 0$ : two distinct real roots; $\Delta = 0$ : equal roots; $\Delta < 0$ : no real roots.
Completed-square form
$ax^2 + bx + c = a(x + h)^2 + k$
$h$
$=\frac{b}{2a}$
$k$
minimum / maximum value of expression
When to use
Finding stationary point and graphing a parabola. For $a > 0$ , min is $k$ ; for $a < 0$ , max is $k$ .

Key Definitions and Keywords — Quadratics

Definitions to memorise and the exact keywords mark schemes credit for quadratics answers — sharpened from recent examiner reports for the 2026 Cambridge International A Level 9709 sitting.

Discriminant
Examiner keyword
The quantity $b^2 - 4ac$ for a quadratic $ax^2 + bx + c$ . Determines the nature of roots.
Completing the square
Examiner keyword
Rewriting $ax^2 + bx + c$ in the form $a(x+h)^2 + k$ . Used to find stationary points, solve equations, and integrate.
Roots
Values of $x$ for which the quadratic equals zero. Where the curve crosses the x-axis.
Vertex
The turning point of a parabola — the stationary point. Coordinates $(-\frac{b}{2a}, c - \frac{b^2}{4a})$ for $y = ax^2 + bx + c$ .

Common Mistakes and Misconceptions — Quadratics

The traps other students keep falling into on quadratics questions — taken from recent Cambridge International A Level 9709 examiner reports and mark schemes — and how to avoid them.

✕Using $\geq$ instead of $>$ for distinct real roots
9709 Examiner Reports 2022-2024
Why it happens
Confusion with 'real roots' (which allows equality).
How to avoid it
Two DISTINCT roots: $\Delta > 0$ . Two real roots (possibly equal): $\Delta \geq 0$ . Equal roots: $\Delta = 0$ .
✕Forgetting to factor coefficient from BOTH x² and x terms
9709 Examiner Reports 2022-2024
Why it happens
Rushing to complete the square.
How to avoid it
For $2x^2 - 12x + 7$ , factor 2 from FIRST TWO terms only: $2(x^2 - 6x) + 7$ . The constant 7 is untouched.
✕Wrong direction for quadratic inequality solution
9709 Examiner Reports 2022-2024
Why it happens
Forgetting parabola shape.
How to avoid it
For $ax^2 + bx + c \geq 0$ with $a > 0$ : solution is OUTSIDE the roots ( $x \leq r_1$ or $x \geq r_2$ ). For $\leq 0$ : solution is BETWEEN the roots.

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

Past paper style quiz

Get a report showing which sub-topics you've nailed and which ones still need work.

Video lesson

Short walkthrough of the concepts students most often get stuck on.

Quadratics — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Quadratics

Short Study Notes in the form of Slides

Detailed Study Notes

What you’ll learn

How it’s examined

1Completing the square (5 marks)

2Using the discriminant (5 marks)

3Simultaneous equations with quadratic (6 marks)

4Quadratic inequality (4 marks)

Quadratic formula

Discriminant

Completed-square form

Discriminant

Completing the square

Roots

Vertex

✕Using ≥\geq≥ instead of >>> for distinct real roots

✕Forgetting to factor coefficient from BOTH x² and x terms

✕Wrong direction for quadratic inequality solution

Practice questions

Past paper style quiz

Assess your understanding

Video lesson

Quadratics — frequently asked questions

✕Using $\geq$ instead of $>$ for distinct real roots