Manipulation of Quadratic Expressions

Edexcel International GCSE Further Pure Mathematics· S2 2A· 15 min read

1. Factorisation of Quadratic Expressions★★☆☆☆⏱ 4 min

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Factorising a quadratic expression of the form $a x^{2} + b x + c$ 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.

📘 Definition

Factorisation of Quadratics

The process of rewriting a quadratic expression $a x^{2} + b x + c$ as the product of two linear factors $(m x + n) (p x + q)$ , where $m, n, p, q$ are integers.

📐 Worked Example

Factorise $2 x^{2} + 7 x + 3$

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: $2 x^{2} + 6 x + x + 3$
3
Factor by grouping: $2 x (x + 3) + 1 (x + 3) = (2 x + 1) (x + 3)$
4
Verify by expanding: $(2 x + 1) (x + 3) = 2 x^{2} + 6 x + x + 3 = 2 x^{2} + 7 x + 3$ , 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.

2. Completing the Square for $ax^2 + bx + c$★★★☆☆⏱ 5 min

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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.

📘 Definition

Completed Square Form

$a (x + p)^{2} + q$

A rearranged form of a quadratic expression $a x^{2} + b x + c$ , where $a$ , $p$ and $q$ are constants, and $a \neq = 0$ .

📐 Worked Example

Rewrite $3 x^{2} - 12 x + 7$ in the form $a (x + p)^{2} + q$

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

3. Using Completed Square Form to Find Quadratic Features★★★☆☆⏱ 4 min

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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.

📘 Definition

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.

4. Common Pitfalls

Wrong move:

Factoring $a$ out of all terms when completing the square for non-monic quadratics, e.g. writing $3 x^{2} - 12 x + 7 = 3 (x^{2} - 4 x + 7)$ instead of $3 (x^{2} - 4 x) + 7$

Why:

This changes the value of the constant term, leading to an incorrect $q$ value and wrong graph features.

Correct move:

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 move:

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:

Misapplying the formula $x = - p$ when the bracket has a positive sign, leading to incorrect graph coordinates.

Correct move:

Set the bracket term equal to zero: $x + 2 = 0 ⟹ x = - 2$ , which is the correct x-coordinate of the vertex.

Wrong move:

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:

The subtracted term is inside the bracket multiplied by $a$ , so failing to scale it leads to an incorrect constant term.

Correct move:

Distribute $a$ to both terms inside the bracket before combining constant terms.

Wrong move:

Assuming $q$ is always the minimum value of the quadratic, regardless of the sign of $a$

Why:

If $a$ is negative, the parabola opens downward, so $q$ is the maximum value, not minimum.

Correct move:

Check the sign of $a$ first: $a > 0$ means $q$ is the minimum value, $a < 0$ means $q$ is the maximum value.

Wrong move:

Skipping the expansion check after factorising a non-monic quadratic

Why:

Small sign errors in factor pairs often lead to incorrect expanded forms that lose marks in exams.

Correct move:

Always expand your factorised result immediately to confirm it matches the original quadratic expression.

5. Quick Reference Cheatsheet

Operation	Steps	Key Output
Factorise $a x^{2} + b x + c$	Calculate $a \times c$ 2. Find factors of $a c$ summing to $b$ 3. Split middle term, factor by grouping	$(m x + n) (p x + q)$
Complete square for $a x^{2} + b x + c$	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

Going deeper

syllabus_documentEdexcel 4PM1 SpecificationOfficial syllabus for this topic

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.