# Manipulation of Quadratic Expressions

> Edexcel International GCSE Further Pure Mathematics · 4PM1
> Source: https://www.owlsprep.com/study/edexcel-igcse-further-pure-math-s2-manipulation-of-quadratic-expressions/

This guide covers core quadratic manipulation skills for Edexcel IGCSE Further Pure Maths (4PM1): factorisation of $ax^2 + bx + c$, completing the square, and using completed square form to identify key quadratic graph features.

**Prerequisites:** [Basic algebraic expansion and simplification](https://www.owlsprep.com/study/edexcel-igcse-maths-algebraic-expansion/); [Understanding of parabola graph shapes](https://www.owlsprep.com/study/edexcel-igcse-fpm-quadratic-graphs-basics/)

## Learning objectives

- Factorise quadratic expressions of the form $ax^2 + bx + c$
- Rewrite $ax^2 + bx + c$ in completed square form $a(x + p)^2 + q$
- Use completed square form to find vertex, maximum/minimum value and line of symmetry of a quadratic graph

## Factorisation of Quadratic Expressions

Factorising a quadratic expression of the form $ax^2 + bx + c$ involves rewriting it as a product of two linear brackets. For monic quadratics ($a=1$), find two numbers that multiply to $c$ and add to $b$. For non-monic quadratics ($a>1$), use the AC method: multiply $a$ and $c$, find two factors of this product that add to $b$, split the middle term, then factor by grouping.

**Factorisation of Quadratics** — The process of rewriting a quadratic expression $ax^2 + bx + c$ as the product of two linear factors $(mx + n)(px + q)$, where $m, n, p, q$ are integers.

**Worked example:** Factorise $2x^2 + 7x + 3$

1. Calculate $a \times c = 2 \times 3 = 6$. Find two numbers that multiply to 6 and add to 7: 6 and 1.
2. Split the middle term using these numbers: $2x^2 + 6x + x + 3$
3. Factor by grouping: $2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)$
4. Verify by expanding: $(2x + 1)(x + 3) = 2x^2 + 6x + x + 3 = 2x^2 +7x +3$, which matches the original expression.

> **Exam tip:** Always expand your factorised result to check for arithmetic errors, especially with negative signs, as these are a common source of lost marks.

*Calculator:* allowed

## Completing the Square for $ax^2 + bx + c$

Completing the square rewrites a quadratic expression in the form $a(x + p)^2 + q$, which reveals key properties of the associated parabola graph. For monic quadratics, halve the coefficient of $x$, square it, then adjust the constant term to maintain equality. For non-monic quadratics, first factor out the coefficient of $x^2$ from the first two terms before completing the square inside the bracket.

**Completed Square Form** — A rearranged form of a quadratic expression $ax^2 + bx + c$, where $a$, $p$ and $q$ are constants, and $a \neq 0$.

*Notation:* $a(x + p)^2 + q$

> **mnemonic**
>
> Factor, Halve, Square, Adjust (F-H-S-A): use this 4-step mnemonic to avoid errors when completing the square for non-monic quadratics.

**Worked example:** Rewrite $3x^2 - 12x + 7$ in the form $a(x + p)^2 + q$

1. Factor out the coefficient of $x^2$ (3) from the first two terms: $3(x^2 - 4x) + 7$
2. Complete the square inside the bracket: half of -4 is -2, square of -2 is 4. Add and subtract 4 inside the bracket: $3[(x^2 -4x +4) -4] +7$
3. Rewrite the perfect square: $3[(x - 2)^2 - 4] +7$
4. Expand the outer bracket and simplify: $3(x - 2)^2 - 12 +7 = 3(x -2)^2 -5$
5. Verify by expanding: $3(x^2 -4x +4) -5 = 3x^2 -12x +12 -5 = 3x^2 -12x +7$, which is correct.

*Calculator:* allowed

## Using Completed Square Form to Find Quadratic Features

Once a quadratic is written in completed square form $a(x + p)^2 + q$, you can directly read off key features of its parabola graph without plotting points or solving equations. These features are tested frequently in 4PM1 exams, often linked to sketching quadratic graphs.

**Key Quadratic Features from Completed Square Form** — For $y = a(x + p)^2 + q$: 1. Line of symmetry is $x = -p$, 2. Vertex (turning point) is at $(-p, q)$, 3. If $a > 0$, $q$ is the minimum value of the quadratic; if $a < 0$, $q$ is the maximum value.

**Worked example:** For the quadratic $y = 2(x - 3)^2 + 5$, state its line of symmetry, vertex coordinates, and minimum value.

1. Match the given form to $a(x + p)^2 + q$: $a=2$, $p=-3$, $q=5$.
2. Line of symmetry is $x = -p = -(-3) = 3$.
3. Vertex coordinates are $(-p, q) = (3, 5)$.
4. Since $a = 2 > 0$, the parabola opens upwards, so the minimum value is $q = 5$.

> **Exam tip:** Always check the sign of $p$ when finding the vertex: set the bracket term equal to zero to confirm the x-coordinate, rather than relying on memorisation to avoid sign errors.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Factoring $a$ out of all terms when completing the square for non-monic quadratics, e.g. writing $3x^2 -12x +7 = 3(x^2 -4x +7)$ instead of $3(x^2 -4x) +7$
  - Why it fails: This changes the value of the constant term, leading to an incorrect $q$ value and wrong graph features.
  - Correct: Only factor $a$ out of the $x^2$ and $x$ terms, leave the constant term outside the bracket before adjusting for the perfect square.
- **Wrong:** Mixing up the sign of the x-coordinate of the vertex, e.g. taking vertex of $y=(x+2)^2 +3$ as (2, 3) instead of (-2, 3)
  - Why it fails: Misapplying the formula $x = -p$ when the bracket has a positive sign, leading to incorrect graph coordinates.
  - Correct: Set the bracket term equal to zero: $x + 2 = 0 \implies x = -2$, which is the correct x-coordinate of the vertex.
- **Wrong:** Forgetting to multiply the subtracted square term by $a$ when expanding the bracket, e.g. writing $3[(x-2)^2 -4] +7 = 3(x-2)^2 -4 +7$ instead of $3(x-2)^2 -12 +7$
  - Why it fails: The subtracted term is inside the bracket multiplied by $a$, so failing to scale it leads to an incorrect constant term.
  - Correct: Distribute $a$ to both terms inside the bracket before combining constant terms.
- **Wrong:** Assuming $q$ is always the minimum value of the quadratic, regardless of the sign of $a$
  - Why it fails: If $a$ is negative, the parabola opens downward, so $q$ is the maximum value, not minimum.
  - Correct: Check the sign of $a$ first: $a>0$ means $q$ is the minimum value, $a<0$ means $q$ is the maximum value.
- **Wrong:** Skipping the expansion check after factorising a non-monic quadratic
  - Why it fails: Small sign errors in factor pairs often lead to incorrect expanded forms that lose marks in exams.
  - Correct: Always expand your factorised result immediately to confirm it matches the original quadratic expression.

## Cheatsheet

| Operation | Steps | Key Output |
| --- | --- | --- |
| Factorise $ax^2 + bx + c$ | 1. Calculate $a \times c$ 2. Find factors of $ac$ summing to $b$ 3. Split middle term, factor by grouping | $(mx + n)(px + q)$ |
| Complete square for $ax^2 + bx + c$ | 1. Factor $a$ from $x^2, x$ terms 2. Halve $x$ coefficient, square, adjust constant 3. Simplify to standard form | $a(x + p)^2 + q$ |
| Find line of symmetry | From $a(x + p)^2 + q$, compute $x = -p$ | Equation of vertical line through vertex |
| Find vertex coordinates | From $a(x + p)^2 + q$, use $(-p, q)$ | Coordinates of turning point |
| Find min/max value | Check sign of $a$: $a>0$ = min $q$, $a<0$ = max $q$ | Extreme value of quadratic |

## What's next

Mastering quadratic manipulation is the foundation for all subsequent quadratic topics in your Edexcel IGCSE Further Pure Maths course. You will use these skills constantly when solving quadratic equations, analyzing the discriminant, and working with symmetric functions of roots in upcoming units. Factorisation and completing the square are also essential for solving quadratic inequalities, sketching parabolas, and simplifying algebraic expressions in later pure maths topics. Make sure you practice these skills regularly with past paper questions to avoid common sign errors and build speed, as they are tested in almost every 4PM1 exam paper, often as the first step in longer multi-mark questions.

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