Simplifying an algebraic expression means collecting like terms and expanding brackets until nothing more can be combined. The three identities, (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b², and (a + b)(a − b) = a² − b², let you expand or factorise these exact patterns instantly, without multiplying term by term. Spotting a match saves time on every algebra question that follows.
Simplifying looks easy until an expression mixes several skills at once: collecting like terms, expanding brackets with the distributive rule, and applying an identity, all in one line. Students who treat each skill separately, checking terms one at a time instead of scanning the whole expression first, lose marks to sign errors and terms left uncombined.
The fix is to work in a fixed order every time: expand brackets first, then collect like terms, watching throughout for whether part of the expression matches one of the three identities below.
Step 1
Identify like terms: same variable, same power
Step 2
Expand brackets: multiply every inside term by the outside term
Step 3
Check for a match to one of the three identities
Step 4
Collect remaining like terms to the simplest form
The three identities
(a + b)² = a² + 2ab + b²(a − b)² = a² − 2ab + b²(a + b)(a − b) = a² − b² (difference of two squares)Simplify 3(2x + 5) − 4(x − 2).
Solution
3(2x + 5) = 6x + 15.−4(x − 2) = −4x + 8.6x + 15 − 4x + 8.(6x − 4x) + (15 + 8) = 2x + 23.Quick check: substitute a simple value such as x = 1 into the original and your final answer. Original: 3(7) − 4(−1) = 21 + 4 = 25. Final: 2(1) + 23 = 25. They match, so the simplification is correct.
Expand and simplify (2x + 3)².
Solution
(a + b)² = a² + 2ab + b² with a = 2x and b = 3.a² = (2x)² = 4x².2ab = 2(2x)(3) = 12x.b² = 3² = 9.(2x + 3)² = 4x² + 12x + 9.Simplify (5x + 2)(5x − 2).
Solution
(a + b)(a − b) = a² − b² with a = 5x and b = 2.a² = (5x)² = 25x².b² = 2² = 4.(5x + 2)(5x − 2) = 25x² − 4. Note there is no middle term at all.The step students get wrong
The most common error is dropping the middle term 2ab when squaring a bracket, writing (a + b)² = a² + b². This is false: (a + b)² means (a + b)(a + b), and every term in the first bracket must multiply every term in the second, giving four products in total, two of which combine into 2ab. The second common error is mixing up the identities: only the plus-then-minus bracket pair, (a + b)(a − b), produces a clean difference with no middle term. Squaring a single bracket, whether (a + b)² or (a − b)², always keeps the middle term.
Simplify 2(x + 1)² − (x + 3)(x − 3).
Solution
(a + b)² = a² + 2ab + b² with a = x, b = 1: (x + 1)² = x² + 2x + 1. Multiply by 2: 2x² + 4x + 2.(a + b)(a − b) = a² − b² with a = x, b = 3: (x + 3)(x − 3) = x² − 9.(2x² + 4x + 2) − (x² − 9).2x² + 4x + 2 − x² + 9.(2x² − x²) + 4x + (2 + 9) = x² + 4x + 11.Exam tip: when subtracting an entire expanded bracket, write it in brackets first and distribute the minus sign in a separate line. Skipping this step, and trying to change every sign in your head while writing, is where most sign errors happen.
How do I know which identity to use?
Look at the two brackets. If they are identical, such as (x + 4)(x + 4) or written as (x + 4)², use (a + b)² = a² + 2ab + b² or (a − b)² = a² − 2ab + b² depending on the sign. If the two brackets have the same two terms but opposite signs in the middle, such as (x + 4)(x − 4), use (a + b)(a − b) = a² − b². If neither pattern matches, expand normally term by term instead of forcing an identity.
What counts as a "like term"?
Two terms are like terms only if they have exactly the same variable raised to exactly the same power. 3x and 5x are like terms. 3x and 3x² are not, because the powers differ, and neither are 3x and 3y, because the variables differ. Only like terms can be added or subtracted together. If deciding which powers match feels shaky once terms like x² and x³ appear inside brackets, review the index laws first.
Do I need to memorise the three identities, or can I always expand manually?
You can always fall back on expanding term by term, and this always gives the correct answer. But recognising the three identities is faster and reduces the chance of a sign error, especially in questions that combine an identity with other algebra, like worked example 4 above. For O-Level, memorising all three well enough to spot them instantly is worth the time. These same identities reappear when you learn to factorise quadratic expressions, so getting comfortable with them here pays off later.
Is a² − b² the same as (a − b)²?
No, and confusing the two is a common mistake. a² − b² factorises as (a + b)(a − b), with no middle term when expanded back out. (a − b)² expands to a² − 2ab + b², which does have a middle term. These come from different starting expressions and are not interchangeable.
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