What is a quadratic equation in the context of Edexcel IGCSE Mathematics?

In Edexcel IGCSE Mathematics, a quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. This type of equation is a fundamental part of the Equations, Formulae, and Identities topic and involves finding the values of x that satisfy the equation.

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

To solve a quadratic equation using the quadratic formula, you apply the formula x = (-b ± √(b^2 - 4ac)) / (2a). This formula provides the solutions for x in the equation ax^2 + bx + c = 0. It's important to calculate the discriminant (b^2 - 4ac) first to determine the nature of the roots, which can be real and distinct, real and repeated, or complex.

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

The discriminant of a quadratic equation, given by b^2 - 4ac, is crucial in determining the nature of the roots. If the discriminant is positive, the equation has two distinct real roots. If it is zero, there is one repeated real root. If the discriminant is negative, the equation has two complex roots. Understanding the discriminant helps in predicting the type of solutions without solving the equation completely.

Can you explain how to factorize a quadratic equation?

Factorizing a quadratic equation involves expressing it in the form (px + q)(rx + s) = 0, where p, q, r, and s are numbers. This method is applicable when the quadratic can be rewritten as a product of two binomials. To factorize, you need to find two numbers that multiply to ac (the product of the coefficient of x^2 and the constant term) and add up to b (the coefficient of x). This technique is often used when the quadratic is easily factorable and provides a straightforward way to find the roots.

What are some common applications of quadratic equations in real life?

Quadratic equations are widely used in various real-life applications, such as calculating projectile motion, optimizing areas and volumes in geometry, and determining profit maximization in economics. In the Edexcel IGCSE curriculum, understanding these applications helps students appreciate the practical significance of quadratic equations beyond theoretical mathematics.

Why is "Stopping after finding one solution" a common mistake in Quadratic Equations ?

Why it happens: Solving as if it's linear. How to avoid it: Quadratic has up to TWO solutions. Always state both: 'or'.

Why is "Trying to factor without rearranging to $= 0$ first" a common mistake in Quadratic Equations ?

Why it happens: Skipping a step. How to avoid it: Always rearrange to standard form ax^2 + bx + c = 0 first. Then factor or use formula.

Why is "Including invalid solutions in physical-context problems" a common mistake in Quadratic Equations ?

Why it happens: Mathematical answers ≠ valid contextually. How to avoid it: After solving, CHECK each solution. Negative lengths, fractional people, etc. — reject. Source: 4MA1/2H May/Jun 2024 — examiner report Q16

Why is "Not writing down the quadratic formula" a common mistake in Quadratic Equations ?

Why it happens: 'Just' applying it. How to avoid it: Edexcel mark scheme awards M1 for QUOTING the formula, separate from substitution. Quote it explicitly.

Equations, Formulae and IdentitiesEdexcel IGCSE Mathematics (4MA1)

Quadratic Equations

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Quadratic Equations Study Notes — Edexcel IGCSE 4MA1 Higher Tier (2026 onwards)

Solve $ax^{2} + bx + c = 0$ via factoring, completing the square, or the quadratic formula. Featured every Higher Tier paper.

What you’ll learn

Mapped to the Cambridge IGCSE 4MA1 syllabus (2026 onwards).

2.7 — Solve quadratic equations by factorising.
2.7 — Solve quadratic equations using the quadratic formula.
2.7 — Use the discriminant to determine the nature of roots.
2.7 — Set up quadratics from real-world problems.

The three methods

Factor (when easy), formula (always works), completing the square (sometimes).

1. Factorising.

Use when the quadratic factors nicely.

Standard form: $ax^{2} + bx + c = 0$ → $(\square x + \square)(\square x + \square) = 0$ .

Each factor zero gives a solution.

Example. $x^{2} - 7x + 12 = 0$ → $(x - 3)(x - 4) = 0$ → $x = 3$ or $x = 4$ .

2. Quadratic formula.

Always works. Memorise:

$x = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$

Example. $2x^{2} + 5x - 7 = 0$ .

$a=2, b=5, c=-7$ .
Discriminant: $25 + 56 = 81$ .
$x = \dfrac{-5 \pm 9}{4} = 1$ or $-3.5$ .

3. Completing the square.

Used for proof and finding turning points. Covered in dedicated subtopic.

Choosing the method:

Situation	Method
Factors nicely	Factorise
Asked for surd answers	Formula
'Show that' / 'Prove that'	Often completing the square
Don't know if factors	Try factor first, then formula

Worked qualitative. When does the formula give you 'no real solutions'?

When the discriminant is negative: $b^{2} - 4ac < 0$ .
The square root of a negative isn't real.
This means the quadratic doesn't cross the $x$ -axis.

Edexcel tip. Quote the formula explicitly. Mark schemes award M1 for the formula, M1 for substitution, A1 for both correct answers.

Factor: when it factors nicely.
Formula: always works.
Memorise the formula — not on Edexcel formula sheet.
Always state BOTH solutions.

The discriminant — what kind of solutions?

$\Delta = b^{2} - 4ac$ . Tells you the nature of the roots.

Definition. Discriminant $\Delta = b^{2} - 4ac$ .

Discriminant	Roots
$\Delta > 0$	Two distinct real roots
$\Delta = 0$	One repeated real root
$\Delta < 0$	No real roots

Why? From the quadratic formula:

$\Delta > 0$ : $\sqrt{\Delta}$ is real and non-zero. $x = (-b \pm$ real $)/2a$ → two distinct.
$\Delta = 0$ : $\sqrt{\Delta} = 0$ . $x = -b/2a$ → one repeated.
$\Delta < 0$ : $\sqrt{\Delta}$ not real. No real solutions.

Worked example. Show that $x^{2} - 4x + 5 = 0$ has no real solutions.

$\Delta = 16 - 20 = -4 < 0$ .
No real roots.

Edexcel applications. Sometimes Edexcel asks 'find values of $k$ for which the equation has real roots'.

Example. For what values of $k$ does $x^{2} - 4x + k = 0$ have real roots?

Need $\Delta \ge 0$ .
$\Delta = 16 - 4k$ .
$16 - 4k \ge 0$ → $k \le 4$ .

Worked qualitative. Why is $\Delta = 0$ called 'one repeated root'?

The formula gives $x = (-b \pm 0)/2a = -b/2a$ .
'Plus zero' and 'minus zero' both give the same answer.
One value, but it counts as a 'double root'.

Edexcel tip. Always state the discriminant and its sign. 'Since $\Delta = -4 < 0$ , no real roots' earns the mark.

$\Delta = b^{2} - 4ac$ .
$\Delta > 0$ : 2 distinct real.
$\Delta = 0$ : 1 repeated.
$\Delta < 0$ : no real.
Used to find ranges of $k$ .

Quadratic word problems

Set up the quadratic, solve, check context.

Common contexts:

Areas (length × width).
Volumes (squared or cubed dimensions).
Numbers (consecutive products).
Projectile motion (height vs time).

Example. Rectangle: length $(x + 3)$ , width $(x - 1)$ . Area $= 32$ .

$(x + 3)(x - 1) = 32$ .
$x^{2} + 2x - 3 = 32$ .
$x^{2} + 2x - 35 = 0$ .
Factor: $(x + 7)(x - 5) = 0$ .
$x = -7$ or $x = 5$ .
Length must be positive: $x = 5$ .

Always reject invalid solutions.

Negative lengths.
Fractional people.
Times before zero.

Worked qualitative. A ball is thrown upward. Its height (m) at time $t$ (s) is $h = -5t^{2} + 20t$ . When does it hit the ground?

$h = 0$ : $-5t^{2} + 20t = 0$ .
Factor: $-5t(t - 4) = 0$ .
$t = 0$ or $t = 4$ .
$t = 0$ is when it was thrown. $t = 4$ is when it lands.

Edexcel tip. When setting up a quadratic from a word problem, always rearrange to the form $ax^{2} + bx + c = 0$ before solving. Mark schemes credit this step.

Set up equation from problem.
Rearrange to $ax^{2} + bx + c = 0$ .
Solve and check context.
Reject invalid solutions.

How it’s examined

Quadratic equations on EVERY Higher Tier paper (4-8 marks). Often combined with word problems, areas, formulae. Examiner reports flag (1) only stating one solution, (2) not quoting the formula, (3) including invalid solutions in word problems.

Sources: Pearson Edexcel International GCSE Mathematics A 4MA1 specification (2026 onwards); 4MA1 Examiner Reports — Jan and May/Jun 2022-2024 series; Past Papers — 4MA1/2H Jan 2024, 4MA1/2H May/Jun 2024. Last reviewed 2026-05-07.

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Step-by-step worked examples — Quadratic Equations

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

1Solve by factorising
Higher• quadratic, factorise
Question
Solve $x^{2} - 7x + 12 = 0$ .
Step-by-step solution
1. Step 1
  Find two numbers that multiply to $12$ and add to $-7$ .
2. Step 2
  Pair: $(-3, -4)$ . Both negative, multiply to $12$ , sum to $-7$ . ✓
3. Step 3
  Factorise: $(x - 3)(x - 4) = 0$ .
4. Step 4
  Each factor zero: $x = 3$ or $x = 4$ .
Answer
$x = 3$ or $x = 4$
2Solve using the quadratic formula
Higher• Adapted from 4MA1/2H Jan 2024 Q15• quadratic formula
Question
Solve $2x^{2} + 5x - 7 = 0$ . Give your answers to $2$ d.p.
Step-by-step solution
1. Step 1
  Identify $a = 2$ , $b = 5$ , $c = -7$ .
2. Step 2
  Apply the formula:
  $x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$
3. Step 3
  Discriminant: $b^{2} - 4ac = 25 - 4(2)(-7) = 25 + 56 = 81$ . $\sqrt{81} = 9$ .
4. Step 4
  Compute: $x = \dfrac{-5 \pm 9}{4}$ .
5. Step 5
  Two solutions: $x = \dfrac{4}{4} = 1$ or $x = \dfrac{-14}{4} = -3.5$ .
Answer
$x = 1$ or $x = -3.5$
Examiner tip
ALWAYS quote the quadratic formula explicitly. The Edexcel formula sheet does NOT include it — you must memorise.
3Solve by factorising (a > 1)
Higher• quadratic, non-monic
Question
Solve $3x^{2} + 11x + 6 = 0$ .
Step-by-step solution
1. Step 1
  AC method: $a \times c = 18$ . Need pair multiplying to $18$ , adding to $11$ .
2. Step 2
  Pair: $(2, 9)$ . Sum $11$ . ✓
3. Step 3
  Split middle term:
  $3x^{2} + 2x + 9x + 6 = 0$
4. Step 4
  Factor by grouping: $x(3x + 2) + 3(3x + 2) = (3x + 2)(x + 3) = 0$ .
5. Step 5
  Solutions: $x = -\dfrac{2}{3}$ or $x = -3$ .
Answer
$x = -\dfrac{2}{3}$ or $x = -3$
4Quadratic that doesn't factor neatly
Higher• quadratic formula
Question
Solve $x^{2} - 4x - 2 = 0$ . Give your answer in surd form.
Step-by-step solution
1. Step 1
  $a = 1$ , $b = -4$ , $c = -2$ . Discriminant: $16 + 8 = 24$ .
2. Step 2
  $\sqrt{24} = 2\sqrt{6}$ .
3. Step 3
  Apply formula:
  $x = \frac{4 \pm 2\sqrt{6}}{2} = 2 \pm \sqrt{6}$
Answer
$x = 2 + \sqrt{6}$ or $x = 2 - \sqrt{6}$
Examiner tip
When asked for SURD FORM, simplify the radical and don't approximate. $\sqrt{24} = 2\sqrt{6}$ , then divide.
5Quadratic from a word problem
Higher• Adapted from 4MA1/2H May/Jun 2024 Q16• quadratic, word problem
Question
A rectangle has length $(x + 3)$ cm and width $(x - 1)$ cm. The area is $32$ cm². Find $x$ .
Step-by-step solution
1. Step 1
  Area = length × width: $(x + 3)(x - 1) = 32$ .
2. Step 2
  Expand: $x^{2} + 2x - 3 = 32$ .
3. Step 3
  Rearrange: $x^{2} + 2x - 35 = 0$ .
4. Step 4
  Factor: pair multiplying to $-35$ , summing to $2$ : $(7, -5)$ . So $(x + 7)(x - 5) = 0$ .
5. Step 5
  Solutions: $x = -7$ or $x = 5$ . Length must be positive, so $x = 5$ .
Answer
$x = 5$
Examiner tip
After solving, REJECT solutions that don't make physical sense. Length $> 0$ , so $x > 1$ here.

Key Formulae — Quadratic Equations

The formulae you need to memorise for quadratic equations on the Pearson Edexcel IGCSE 4MA1 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 $ax^{2} + bx + c = 0$
$b^{2} - 4ac$
discriminant
When to use
When the quadratic doesn't factor over integers, OR when surd-form solutions are required.
Example
$2x^{2} + 5x - 7 = 0$ : $a=2, b=5, c=-7$ . Discriminant $= 81$ . $x = (-5 \pm 9)/4 = 1$ or $-3.5$ .
Discriminant
$\Delta = b^{2} - 4ac$
$a, b, c$
coefficients
When to use
To determine the NATURE of solutions: $\Delta > 0$ two real, $\Delta = 0$ one repeated, $\Delta < 0$ no real.
Example
$x^{2} - 4x + 5 = 0$ : $\Delta = 16 - 20 = -4 < 0$ → no real solutions.

Key Definitions and Keywords — Quadratic Equations

Definitions to memorise and the exact keywords mark schemes credit for quadratic equations answers — sharpened from recent examiner reports for the 2026 Pearson Edexcel IGCSE 4MA1 sitting.

Quadratic equation
Examiner keyword
An equation of the form $ax^{2} + bx + c = 0$ where $a \ne 0$ .
Example
$x^{2} - 5x + 6 = 0$ .
Discriminant
The expression $b^{2} - 4ac$ . Tells you the number and type of real solutions.
Example
$\Delta = 25 - 4(2)(3) = 1 > 0$ → two real solutions.

Common Mistakes and Misconceptions — Quadratic Equations

The traps other students keep falling into on quadratic equations questions — taken from recent Pearson Edexcel IGCSE 4MA1 examiner reports and mark schemes — and how to avoid them.

✕Stopping after finding one solution
Why it happens
Solving as if it's linear.
How to avoid it
Quadratic has up to TWO solutions. Always state both: 'or'.
✕Trying to factor without rearranging to $= 0$ first
Why it happens
Skipping a step.
How to avoid it
Always rearrange to standard form $ax^{2} + bx + c = 0$ first. Then factor or use formula.
✕Including invalid solutions in physical-context problems
4MA1/2H May/Jun 2024 — examiner report Q16
Why it happens
Mathematical answers ≠ valid contextually.
How to avoid it
After solving, CHECK each solution. Negative lengths, fractional people, etc. — reject.
✕Not writing down the quadratic formula
Why it happens
'Just' applying it.
How to avoid it
Edexcel mark scheme awards M1 for QUOTING the formula, separate from substitution. Quote it explicitly.

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.

Quadratic Equations — frequently asked questions

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

Quadratic Equations

Short Study Notes in the form of Slides

Detailed Study Notes

What you’ll learn

How it’s examined

1Solve by factorising

2Solve using the quadratic formula

3Solve by factorising (a > 1)

4Quadratic that doesn't factor neatly

5Quadratic from a word problem

Quadratic formula

Discriminant

Quadratic equation

Discriminant

✕Stopping after finding one solution

✕Trying to factor without rearranging to =0= 0=0 first

✕Including invalid solutions in physical-context problems

✕Not writing down the quadratic formula

Practice questions

Past paper style quiz

Assess your understanding

Video lesson

Quadratic Equations — frequently asked questions

✕Trying to factor without rearranging to $= 0$ first