Expanding and Simplifying Expressions: KS3 Maths

What does expanding a bracket mean?

Expanding a bracket means multiplying the term outside the bracket by every term inside it. The bracket is removed; the multiplication is applied.

Rule: a(b + c) = ab + ac

Examples:

• 3(x + 5) = 3x + 15
• 4(2y − 3) = 8y − 12
• −2(x + 7) = −2x − 14 (multiply each term by −2)
• x(x + 4) = x² + 4x

Worked step-by-step: Expand 5(3a − 2).

1. Multiply 5 by 3a: 5 × 3a = 15a.
2. Multiply 5 by −2: 5 × (−2) = −10.
3. Result: 15a − 10.

What does simplifying mean, and what are like terms?

Like terms are terms that contain exactly the same variable(s) raised to the same power(s). Only like terms can be collected (added or subtracted).

Simplifying: collect all x-terms, all y-terms, all constant terms separately.

Example: Simplify 5x + 3 − 2x + 7.

1. x-terms: 5x − 2x = 3x.
2. Constants: 3 + 7 = 10.
3. Result: 3x + 10.

How do you expand and simplify in one expression?

When an expression has two or more brackets, expand each one first, then collect like terms.

Worked example: Expand and simplify 3(x + 4) + 2(x − 1).

1. Expand the first bracket: 3(x + 4) = 3x + 12.
2. Expand the second bracket: 2(x − 1) = 2x − 2.
3. Write all terms together: 3x + 12 + 2x − 2.
4. Collect like terms: (3x + 2x) + (12 − 2) = 5x + 10.

Worked example: Expand and simplify 4(2a + 3) − 3(a − 5).

1. 4(2a + 3) = 8a + 12.
2. −3(a − 5) = −3a + 15. (The negative multiplies both terms inside.)
3. 8a + 12 − 3a + 15.
4. Collect: (8a − 3a) + (12 + 15) = 5a + 27.

How do you expand and simplify with different variables?

Keep each variable type separate when collecting.

Worked example: Expand and simplify 2(3x + y) + 4(x − 2y).

1. 2(3x + y) = 6x + 2y.
2. 4(x − 2y) = 4x − 8y.
3. 6x + 2y + 4x − 8y.
4. x-terms: 6x + 4x = 10x. y-terms: 2y − 8y = −6y.
5. Result: 10x − 6y.

Worked example: Expand and simplify 3(a² + 2a) − a(a − 4).

1. 3(a² + 2a) = 3a² + 6a.
2. a(a − 4) = a² − 4a.
3. 3a² + 6a − (a² − 4a) = 3a² + 6a − a² + 4a.
4. a² terms: 3a² − a² = 2a². a-terms: 6a + 4a = 10a.
5. Result: 2a² + 10a.

What mistakes should you avoid?

Mistake 1 — Forgetting to multiply every term inside the bracket. In 3(x + 4), both x and 4 must be multiplied by 3. Writing just 3x + 4 is wrong.

Mistake 2 — Sign errors with a negative multiplier. In −2(x − 5), multiply both terms: −2 × x = −2x and −2 × (−5) = +10. The result is −2x + 10, not −2x − 10.

Mistake 3 — Collecting unlike terms. 3x + 7 cannot be simplified further; 3x and 7 are unlike terms. Do not write "10x".

Mistake 4 — Skipping the expand step. Attempting to simplify 3(x + 4) + 2(x + 1) by treating the whole expression as 5(x + 5) is incorrect — you must expand each bracket separately before adding.

Frequently asked questions

Can I simplify before expanding?

No. Brackets must be expanded first. Before expanding, the terms inside a bracket are a single packaged group — you cannot combine terms from different brackets or reach inside a bracket to collect terms with outside terms.

What if a bracket is multiplied by a negative number?

The negative sign multiplies every term inside the bracket. Be especially careful when the bracket contains a subtraction: −3(x − 4) = −3x + 12, NOT −3x − 12. A good check is to verify the number of terms: a bracket with two terms, multiplied out, always produces two terms (before simplification).

How do I check my answer?

Substitute a simple value (such as x = 1 or x = 2) into both the original expression and your simplified answer. If both give the same result, the expansion is likely correct. For example, checking 3(x + 4) + 2(x − 1) = 5x + 10 with x = 2: original = 3(6) + 2(1) = 18 + 2 = 20; simplified = 5(2) + 10 = 20. ✓

What comes next after expanding single brackets?

At GCSE you will expand two brackets multiplied together, such as (x + 3)(x + 2) = x² + 5x + 6. This is called expanding a pair of brackets or using FOIL/grid method. The process is similar but each term in the first bracket multiplies every term in the second. At KS3, single bracket expansion is the foundation that GCSE double-bracket expansion builds on.

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