How to Expand Single Brackets: IGCSE and GCSE Maths

Learning how to expand single brackets is one of the most important foundational skills in IGCSE and GCSE Maths. You use the distributive law to multiply the term outside the bracket by every term inside, keeping the signs correct throughout. This page covers positive and negative multipliers, variable multipliers such as x(x+3), brackets with several terms, and expanding then collecting like terms. Each type comes with fully worked examples and step-by-step solutions. When you are ready, use the randomly generated, auto-marked practice questions below to build speed and accuracy before your Edexcel IGCSE exam.

Prior Knowledge

You should be comfortable with collecting like terms before starting. The sign rules for multiplication are also important; they are in the section below.

⇓ Jump to Practice Questions ⇓

What does "expand a bracket" mean?

To expand a bracket means to remove the parentheses by multiplying the term outside by every term inside. This is called the distributive law:

\[ k(ax + b) = k \times ax + k \times b \]

If the outside term is negative, it flips the sign of every term inside:

\[ -(ax - b) = -ax + b \]
⚠️ Mind the minus.
A negative outside the bracket changes the sign of every term inside, not just the first. This is the most common single-bracket mistake in IGCSE exams.

Sign rules (for multiplication)

(+) × (+) = (+)
(+) × (−) = (−)
(−) × (+) = (−)
(−) × (−) = (+)
The Distributive Law
\(\displaystyle k(a + b) = ka + kb\)

Step-by-Step Method

  1. Identify the outside multiplier. Is it positive, negative, a number, or a variable?
  2. Multiply it by the first term inside. Keep the sign correct.
  3. Multiply it by every remaining term inside, one at a time.
  4. Write out the expanded expression. Do not skip lines in your working.
  5. Collect like terms if any are present.

Core Ideas

Every term gets multiplied
The outside factor multiplies all terms inside; missing even one term costs the mark.
Negative flips all signs
A negative multiplier changes every sign inside. \(-(x-5)\) becomes \(-x+5\), not \(-x-5\).
Variable multipliers give x²
\(x(x+3) = x^2 + 3x\). The variable outside multiplied by \(x\) inside gives a squared term.
Collect like terms last
Expand fully first, then simplify. Never try to do both at once; that is where sign errors creep in.

Worked Examples

💡 Example 1: Positive multiplier

Expand \(4(x + 3)\)

\[ \begin{array}{rcl} 4(x+3) &=& 4 \times x + 4 \times 3 \\ &=& 4x + 12 \end{array} \]
What's happening?
  • Multiply 4 by \(x\): gives \(4x\).
  • Multiply 4 by 3: gives \(12\).
  • Both terms positive: no sign issues.

💡 Example 2: Negative multiplier

Expand \(-3(2x - 5)\)

\[ \begin{array}{rcl} -3(2x-5) &=& (-3)(2x) + (-3)(-5) \\ &=& -6x + 15 \end{array} \]
What's happening?
  • \(-3 \times 2x = -6x\) (neg × pos = neg).
  • \(-3 \times -5 = +15\) (neg × neg = pos).
  • The minus inside flips to a plus.

💡 Example 3: Variable multiplier

Expand \(x(x + 5)\)

\[ \begin{array}{rcl} x(x+5) &=& x \times x + x \times 5 \\ &=& x^2 + 5x \end{array} \]
What's happening?
  • \(x \times x = x^2\) (variable squared).
  • \(x \times 5 = 5x\).
  • Two unlike terms: cannot simplify further.

💡 Example 4: Three terms inside

Expand \(3(2x - y + 4)\)

\[ \begin{array}{rcl} 3(2x-y+4) &=& 3(2x) + 3(-y) + 3(4) \\ &=& 6x - 3y + 12 \end{array} \]
What's happening?
  • Three terms inside: multiply each separately.
  • \(3 \times (-y) = -3y\).
  • All three unlike terms: the result has three terms.

💡 Example 5: Implied minus one

Expand and simplify \(7 - (2x - 5)\)

\[ \begin{array}{rcl} 7 - (2x-5) &=& 7 - 2x + 5 \\ &=& -2x + 12 \end{array} \]
What's happening?
  • The lone \(-\) means \(-1 \times\) everything inside.
  • \(-1 \times 2x = -2x\).
  • \(-1 \times -5 = +5\), then \(7+5=12\).

💡 Example 6: Expand and collect

Expand and simplify \(3(x+5) + 2(x-1)\)

\[ \begin{array}{rcl} 3(x+5) + 2(x-1) &=& 3x + 15 + 2x - 2 \\ &=& 5x + 13 \end{array} \]
What's happening?
  • Expand each bracket separately first.
  • Then collect like terms: \(3x+2x=5x\), \(15-2=13\).
  • Always expand before collecting.

💡 Example 7: Number and variable multiplier

Expand \(2x(3x - y)\)

\[ \begin{array}{rcl} 2x(3x - y) &=& 2x \times 3x + 2x \times (-y) \\ &=& 6x^2 - 2xy \end{array} \]
What's happening?
  • \(2x \times 3x = 6x^2\): multiply the numbers, then \(x \times x = x^2\).
  • \(2x \times (-y) = -2xy\): a cross product of two different letters.
  • The two terms are unlike, so it does not simplify further.

🔑 Key Points

  • Every term: the outside multiplier hits every single term inside, no exceptions.
  • Sign rules: same signs give positive, different signs give negative.
  • Variable multipliers: \(x \times x = x^2\); remember to square the variable.
  • Three-term brackets: expand all three terms, then collect like terms at the end.
  • Work neatly: one line per step, aligned equals signs; examiners award method marks.

⚠️ Common Pitfalls

  • Only multiplying the first term inside and forgetting the rest.
  • With \(-(ax - b)\): the minus flips the inner minus too, giving \(-ax + b\), not \(-ax - b\).
  • Trying to collect like terms before fully expanding both brackets.
  • Writing \(x(x+3) = x^2+3\); forgetting to multiply \(x \times 3 = 3x\), not just 3.
⇓ Jump to Practice Questions ⇓

Ready for the next step? Expanding double brackets builds directly on this skill.

Next up: Double Brackets →

Expanding Single Brackets: Practice Rooms

These randomly generated, auto-marked questions show you how to expand single brackets, from number and variable multipliers through to expanding and collecting like terms, ready for your Edexcel IGCSE Maths exam. Type answers in simplified form (for example 4x+12, -2x+7, 2ab+3a, x^2+5x); use ^ for powers, or the x² button. The order of the terms does not matter.

✓ Correct 0
✗ Re-attempts 0
🔥 Streak 0
🏆 Best 0

Difficulty rises down each column. Answers must be fully expanded and simplified; the order of terms does not matter.

Video guide on how to expand single brackets

This YouTube video provides a comprehensive explanation of expanding single brackets. The video demonstrates the step-by-step process with clear examples, allowing you to understand and visualize the expansion technique. Take the time to watch the video, and you’ll gain a solid foundation in expanding single brackets in algebraic expressions.