Algebraic Formulae & Identities, Explained the Simple Way
Every identity from (a+b)² to a³+b³+c³−3abc — with derivations, solved examples, memory tricks, common mistakes, and exam-ready practice. Built for NCERT, CBSE and State Board (SEBA/Assam Board) students of Classes 8–10.
What Exactly Is an Algebraic Identity?
Let's clear this up before anything else, because most confusion in algebra starts right here.
An algebraic equation is true only for certain values of the variable. For example, x + 2 = 5 is true only when x = 3.
An algebraic identity, on the other hand, is true for every possible value of the variables involved. For example, (a+b)² = a² + 2ab + b² is true whether a = 2, a = −7, a = 0.5, or anything else — you can plug in any numbers and both sides will always match.
Think of an identity as a verified shortcut — one that mathematicians have already proven works every single time, so you don't have to multiply everything out longhand again and again.
That's the entire point of this guide: once you trust that an identity always works, you can use it to expand expressions in seconds, factorise tricky polynomials, and even do lightning-fast mental maths (like squaring 105 without a calculator).
The Geometric Proof of (a+b)²
Drag the sliders below and watch why (a+b)² is never just a² + b². The square's area is built from four smaller pieces — and that's the whole secret of the identity.
The big square of side (a+b) is made of one a×a square, one b×b square, and two a×b rectangles. Add their areas — that's exactly the identity!
📊 Master Formula Chart — All Identities at a Glance
Your one-stop revision sheet. Bookmark this page and scroll back here the night before your exam.
| # | Identity | Formula | Best used for |
|---|---|---|---|
| 1 | Square of a sum | (a+b)² = a²+2ab+b² | Expanding, mental squares |
| 2 | Square of a difference | (a−b)² = a²−2ab+b² | Expanding, mental squares |
| 3 | Difference of squares | (a+b)(a−b) = a²−b² | Fast multiplication, factorising |
| 4 | Cube of a sum | (a+b)³ = a³+3a²b+3ab²+b³ | Expanding cubes |
| 5 | Cube of a difference | (a−b)³ = a³−3a²b+3ab²−b³ | Expanding cubes |
| 6 | Sum of cubes | a³+b³ = (a+b)(a²−ab+b²) | Factorising |
| 7 | Difference of cubes | a³−b³ = (a−b)(a²+ab+b²) | Factorising |
| 8 | Square of a trinomial | (a+b+c)² = a²+b²+c²+2ab+2bc+2ca | 3-variable expansion |
| 9 | Sum of three cubes | a³+b³+c³−3abc = (a+b+c)(a²+b²+c²−ab−bc−ca) | Factorising, special case a+b+c=0 |
| 10 | Product of two binomials | (x+a)(x+b) = x²+(a+b)x+ab | Quick multiplication, quadratic forming |
Every Identity, Fully Explained
For each formula: what it means, how it's derived, worked examples, a memory trick, the mistakes examiners see most often, and a quick check for you to try.
Square of a Sum
In Plain Language
When you square the sum of two terms, you don't just square each term separately — you also pick up a "cross term" of double their product. This is the single most-used identity in all of school algebra, so make this one automatic.
Derivation
(a+b)² = (a+b) × (a+b)
= a(a+b) + b(a+b)
= a² + ab + ab + b²
= a² + 2ab + b²
Notice the two ab terms are identical, so they combine into 2ab — that's where the "double product" comes from.
Solved Examples
Here a = 3x, b = 4y.
a² = 9x², 2ab = 2(3x)(4y) = 24xy, b² = 16y².
Answer: 9x² + 24xy + 16y²
Write 105 = 100 + 5. Then 105² = (100+5)² = 100² + 2(100)(5) + 5² = 10000 + 1000 + 25.
Answer: 11025
From the identity, a² + b² = (a+b)² − 2ab = 10² − 2(21) = 100 − 42.
Answer: 58
Memory Trick
Say it like a rhythm: "First squared, plus twice the product, plus second squared." Or remember FIRST² + 2(FIRST·SECOND) + SECOND² — the pattern never changes no matter what a and b are.
Common Mistakes
- (a+b)² = a² + b² — Wrong! You must include the middle term 2ab. This is the #1 mistake examiners report.
- Forgetting to double the product: writing a² + ab + b² instead of a² + 2ab + b².
- Sign error when a or b is itself negative — always substitute the full term including its sign before squaring.
Quick Check: Expand (2p + 5q)²
Square of a Difference
In Plain Language
Almost identical to Identity 1, except the cross term is now subtracted. The squared terms a² and b² stay positive (squares are never negative!) — only the middle term changes sign.
Derivation
(a−b)² = (a−b) × (a−b)
= a(a−b) − b(a−b)
= a² − ab − ab + b²
= a² − 2ab + b²
Two negative ab terms combine to give −2ab, while (−b)×(−b) = +b² stays positive.
Solved Examples
a = 5p, b = 3q → a² = 25p², 2ab = 30pq, b² = 9q².
Answer: 25p² − 30pq + 9q²
Write 98 = 100 − 2. Then 98² = (100−2)² = 10000 − 2(100)(2) + 4 = 10000 − 400 + 4.
Answer: 9604
x² + y² = (x−y)² + 2xy = 36 + 14.
Answer: 50
Memory Trick
Same chant as before, just flip one sign: "First squared, minus twice the product, plus second squared." The b² is ALWAYS positive — students often wrongly make it negative because the formula has a minus sign in it.
Common Mistakes
- Writing a² − 2ab − b² instead of a² − 2ab + b² — the last term is always added, never subtracted.
- Treating (a−b)² as a² − b² (that's actually a completely different identity — see Identity 3 below!).
- Sign confusion when a or b is negative, e.g., for (x − (−3))², simplify inside the bracket first to (x+3)² before applying the formula.
Quick Check: Expand (7m − 2n)²
Difference of Squares
In Plain Language
This is the "magic disappearing" identity. When you multiply a sum and a difference of the same two terms, the middle terms cancel out completely, leaving only a difference of two squares. It's the fastest identity for mental multiplication.
Derivation
(a+b)(a−b) = a(a−b) + b(a−b)
= a² − ab + ab − b²
= a² − b²
The −ab and +ab are exact opposites, so they cancel — that's the entire trick.
Solved Examples
Write as (100+4)(100−4) = 100² − 4² = 10000 − 16.
Answer: 9984
49x² = (7x)², 81y² = (9y)² → (7x)² − (9y)².
Answer: (7x + 9y)(7x − 9y)
Directly apply the identity with a = x, b = 7.
Answer: x² − 49
Memory Trick
"Same, same, opposite signs → squares subtract." Whenever you see the exact same two terms, once added and once subtracted, skip the full expansion entirely and jump straight to a² − b². This single identity can save you 30+ seconds per question in MCQs.
Common Mistakes
- Applying this identity when the terms aren't actually identical, e.g. wrongly simplifying (x+3)(x−5) as x²−15 — this identity ONLY works when both brackets have the same two terms.
- Writing a² + b² instead of a² − b² — the result is always a subtraction.
- Forgetting to take the square root correctly while factorising, e.g. mistaking 49 for 7² but writing it as 14 by mistake.
Quick Check: Simplify (9a + 2b)(9a − 2b)
Cube of a Sum
In Plain Language
Now we're cubing instead of squaring, so there are four terms instead of three. Notice the pattern in the coefficients: 1, 3, 3, 1 — the powers of a go down (3,2,1,0) while powers of b go up (0,1,2,3) as you move left to right.
Derivation
(a+b)³ = (a+b)² × (a+b)
= (a² + 2ab + b²)(a+b)
= a³ + a²b + 2a²b + 2ab² + ab² + b³
= a³ + 3a²b + 3ab² + b³
There's also a handy shortcut form: (a+b)³ = a³ + b³ + 3ab(a+b) — useful when you already know a+b and ab.
Solved Examples
a = x, b = 2 → x³ + 3x²(2) + 3x(4) + 8 = x³ + 6x² + 12x + 8.
Answer: x³ + 6x² + 12x + 8
Write 11 = 10 + 1. (10+1)³ = 1000 + 3(100)(1) + 3(10)(1) + 1 = 1000+300+30+1.
Answer: 1331
Use a³+b³ = (a+b)³ − 3ab(a+b) = 5³ − 3(6)(5) = 125 − 90.
Answer: 35
Memory Trick
Remember the coefficient pattern 1-3-3-1 (it's actually Pascal's Triangle row for power 3!). Pair that with the rule: a's power falls, b's power rises, every term's powers add up to 3.
Common Mistakes
- The biggest one: writing (a+b)³ = a³ + b³ — this is true ONLY for the special case in Identity 9 (when a+b+c=0), never in general. Always include the two middle terms.
- Forgetting the coefficient 3 on the middle terms: writing a²b instead of 3a²b.
- Mixing up which term gets squared — it's 3a²b (a squared) then 3ab² (b squared), not the reverse.
Quick Check: Expand (y + 3)³
Cube of a Difference
In Plain Language
The same 1-3-3-1 pattern as before, but now the signs alternate: plus, minus, plus, minus. Watch this one carefully — it's the formula students lose the most marks on due to sign slips.
Derivation
(a−b)³ = (a−b)² × (a−b)
= (a² − 2ab + b²)(a−b)
= a³ − a²b − 2a²b + 2ab² + ab² − b³
= a³ − 3a²b + 3ab² − b³
Shortcut form: (a−b)³ = a³ − b³ − 3ab(a−b).
Solved Examples
a = x, b = 3 → x³ − 3x²(3) + 3x(9) − 27 = x³ − 9x² + 27x − 27.
Answer: x³ − 9x² + 27x − 27
Write 9 = 10 − 1. (10−1)³ = 1000 − 3(100)(1) + 3(10)(1) − 1 = 1000−300+30−1.
Answer: 729
Use a³−b³ = (a−b)³ + 3ab(a−b) = 4³ + 3(5)(4) = 64 + 60.
Answer: 124
Memory Trick
Use the phrase "PLUS MINUS PLUS MINUS" like a heartbeat as you write each of the four terms left to right. If you remember Identity 4 correctly, just alternate the signs starting with plus.
Common Mistakes
- Getting the sign pattern wrong — it is +, −, +, −, never −, +, −, + or all negative.
- Writing the last term as +b³ instead of −b³ — odd powers of a negative number stay negative.
- Confusing this with a³ − b³ (Identity 7) — that one has only TWO terms inside one bracket, this one expands into FOUR terms.
Quick Check: Expand (2m − 1)³
Sum of Cubes (Factorised Form)
In Plain Language
This goes the opposite direction from Identity 4 — instead of expanding a cube, we're factorising a sum of two cubes back into a product. The second bracket looks similar to a perfect square but has a minus in the middle instead of a plus.
Derivation
Start from the cube identity: (a+b)³ = a³ + 3a²b + 3ab² + b³ = a³ + b³ + 3ab(a+b)
⇒ a³ + b³ = (a+b)³ − 3ab(a+b)
= (a+b)[(a+b)² − 3ab]
= (a+b)[a² + 2ab + b² − 3ab]
= (a+b)(a² − ab + b²)
Solved Examples
8x³ = (2x)³, 27y³ = (3y)³ → a = 2x, b = 3y.
Answer: (2x + 3y)(4x² − 6xy + 9y²)
x²+y² = (x+y)²−2xy = 16−6 = 10. So a²−ab+b² = 10−3 = 7.
x³+y³ = (x+y)(7) = 4 × 7.
Answer: 28
125 = 5³ → a-term is 5, b-term is a.
Answer: (5 + a)(25 − 5a + a²)
Memory Trick
Say: "SAME sign outside, ALWAYS minus in the middle, ALWAYS plus at the end." The first bracket (a+b) keeps the same sign as the original problem, and the second bracket a²−ab+b² ALWAYS has a minus in the middle — no matter what.
Common Mistakes
- Writing the second bracket as a² + ab + b² instead of a² − ab + b² — this is the most common slip and it's the opposite identity (Identity 7's bracket).
- Forgetting to express numbers as perfect cubes first — always check if 8, 27, 64, 125, 1000 etc. are cubes before applying the formula.
- Confusing this with (a+b)³ — remember, this identity factorises, it doesn't expand.
Quick Check: Factorise 1 + 64p³
Difference of Cubes (Factorised Form)
In Plain Language
The mirror image of Identity 6. This time the first bracket carries the minus sign, but — and this trips up almost everyone — the second bracket flips to a PLUS in the middle.
Derivation
Start from: (a−b)³ = a³ − 3a²b + 3ab² − b³ = a³ − b³ − 3ab(a−b)
⇒ a³ − b³ = (a−b)³ + 3ab(a−b)
= (a−b)[(a−b)² + 3ab]
= (a−b)[a² − 2ab + b² + 3ab]
= (a−b)(a² + ab + b²)
Solved Examples
64x³ = (4x)³, 1 = 1³ → a = 4x, b = 1.
Answer: (4x − 1)(16x² + 4x + 1)
x²+y² = (x−y)²+2xy = 9+8 = 17. So a²+ab+b² = 17+4 = 21.
x³−y³ = (x−y)(21) = 3 × 21.
Answer: 63
1000 = 10³ → a-term is 10, b-term is a.
Answer: (10 − a)(100 + 10a + a²)
Memory Trick
Pair the two cube identities side by side: for SUM of cubes, the second bracket is minus in the middle; for DIFFERENCE of cubes, the second bracket is plus in the middle. The signs literally swap between Identity 6 and 7 — learn them together, never alone.
Common Mistakes
- Writing a² − ab + b² instead of a² + ab + b² in the second bracket — students often just copy Identity 6's bracket by habit.
- Mixing up a and b when the difference is written "backwards" — e.g. for 1 − 8x³, set a = 1, b = 2x, don't flip them.
- Stopping after finding (a−b) and forgetting to write the trinomial bracket — both factors are needed for full marks.
Quick Check: Factorise 27p³ − 8
Square of a Trinomial
In Plain Language
Now we have three terms instead of two. You square each one individually, then add the double-product of every possible pair — that's ab, bc, and ca. Miss even one pair and your answer will be wrong.
Derivation
Treat (b+c) as a single term and apply Identity 1: (a+(b+c))² = a² + 2a(b+c) + (b+c)²
= a² + 2ab + 2ac + b² + 2bc + c²
= a² + b² + c² + 2ab + 2bc + 2ca
Solved Examples
a=x, b=y, c=2 → x²+y²+4 + 2xy + 4y + 4x.
Answer: x²+y²+4+2xy+4x+4y
a²+b²+c² = (a+b+c)² − 2(ab+bc+ca) = 81 − 52.
Answer: 29
Treat as a=2x, b=−y, c=3 → 4x²+y²+9 + 2(2x)(−y) + 2(−y)(3) + 2(3)(2x).
Answer: 4x²+y²+9−4xy−6y+12x
Memory Trick
Picture a triangle with corners a, b, c. Square each corner, then draw all 3 connecting edges and double each one: 2ab + 2bc + 2ca. Three corners, three edges — nothing left out.
Common Mistakes
- The most common error: only writing 2ab and forgetting 2bc and 2ca — all three cross-pairs are required.
- Losing track of signs when b or c is negative — substitute the term with its sign first, then square/multiply.
- Writing 2(a+b+c) instead of the three separate double-products — they cannot be combined that simply.
Quick Check: Expand (p + 2q − r)²
Sum of Three Cubes (Special Identity)
In Plain Language
This looks intimidating but it's mostly used for one famous special case: if a+b+c = 0, then a³+b³+c³ = 3abc. That single shortcut shows up again and again in HOTS and competitive exam questions.
Derivation (of the special case)
From the identity: a³+b³+c³−3abc = (a+b+c)(a²+b²+c²−ab−bc−ca)
If a+b+c = 0, the right-hand side becomes 0 (since anything multiplied by 0 is 0):
⇒ a³+b³+c³ − 3abc = 0
⇒ a³+b³+c³ = 3abc
Solved Examples
a³+b³+c³ = 1+8−27 = −18. And 3abc = 3(1)(2)(−3) = −18.
Both sides match: −18 = −18 ✓
Let a=x, b=y, c=−3 so a+b+c=0 ⇒ x³+y³+(−3)³ = 3·x·y·(−3) ⇒ x³+y³−27 = −9xy.
Answer: x³+y³−27 = −9xy
Write as a³+b³+2³−3(a)(b)(2) → factor using the main identity with c=2.
Answer: (a+b+2)(a²+b²+4−ab−2b−2a)
Memory Trick
"Zero sum, triple product." Whenever a question gives you three terms that add up to zero and asks for the sum of their cubes, your alarm bell should ring instantly: the answer is simply 3 times their product.
Common Mistakes
- Trying to apply the "a+b+c=0" shortcut without checking that the three terms actually sum to zero — always verify first.
- Forgetting the minus signs in the second bracket — it's a²+b²+c²−ab−bc−ca, all three cross-products subtracted.
- Confusing this with Identity 8 — Identity 8 squares a trinomial; this one is about cubes and includes an extra −3abc term.
Quick Check: If x+y+z=0, what does x³+y³+z³ equal?
Product of Two Binomials
In Plain Language
This identity is the bridge between algebra and quadratic equations. It tells you that whenever you multiply two binomials that share the same variable x, the middle coefficient is the SUM of the two constants, and the last term is their PRODUCT.
Derivation
(x+a)(x+b) = x(x+b) + a(x+b)
= x² + bx + ax + ab
= x² + (a+b)x + ab
Solved Examples
a=3, b=5 → sum=8, product=15.
Answer: x² + 8x + 15
a=−4, b=6 → sum=2, product=−24.
Answer: x² + 2x − 24
Write as (100+2)(100+8) with x=100, a=2, b=8 → 100² + (2+8)(100) + (2)(8) = 10000+1000+16.
Answer: 11016
Memory Trick
"Sum in the middle, product at the end." This is also exactly how you factorise quadratics in reverse — given x²+8x+15, find two numbers that ADD to 8 and MULTIPLY to 15 (here, 3 and 5), and you've cracked it.
Common Mistakes
- Forgetting signs of a and b when one is negative — always carry the sign into BOTH the sum and the product, e.g. for (x−4)(x+6), a is −4, not 4.
- Confusing this with Identity 1 — this identity has two DIFFERENT constants a and b, not the same term squared.
- Adding instead of multiplying for the last term, or vice versa — remember, middle = sum, last = product.
Quick Check: Expand (x−7)(x−2)
🔄 Flip Cards — Test Yourself Before You Flip
Tap any card to flip it. Try to say the expansion out loud BEFORE you flip — that's how real recall is built, not by re-reading.
🌍 Where These Identities Actually Show Up in Real Life
Algebraic identities aren't just exam tools — they quietly power everyday calculations and entire careers.
Fast Mental Maths While Shopping
Calculating a 95-rupee item × 12 quantity, or squaring numbers near 100 (like 98² or 103²), becomes instant using the difference-of-squares and square identities — no calculator needed.
Land & Farm Area Calculations
If a square plot's side increases by a few metres, (a+b)² instantly tells you the new total area without re-measuring — genuinely useful for farmers and surveyors estimating land.
Construction & Engineering
Architects and civil engineers use squared and cubed expressions constantly when calculating areas, volumes of concrete, and load distributions on box-shaped structures.
Computer Science & Cryptography
The difference-of-squares identity is a key trick in number theory used in factorisation algorithms, which underpin parts of modern cryptography and secure communication.
Physics Formulas
Equations like kinetic energy, electrical power (P=I²R), and projectile motion frequently involve squared terms — recognising (a+b)² patterns helps you simplify physics derivations faster.
Higher Studies & Competitive Exams
Every one of these identities reappears in Class 11–12 algebra, JEE/NEET-level problems, and aptitude sections of competitive and government exams — master them now, reuse them for years.
🎯 Exam Tips From the Examiner's Chair
These small habits are worth real marks in CBSE, SEBA, and other State Board exams.
- STEP MARKS Always write the identity itself as your first line before substituting values — most marking schemes award a mark just for stating the correct formula, even if a later step has a small slip.
- SPEED For "find the value of 105² / 998² / 95×105"-type MCQs, identify the nearby round number instantly and apply Identity 1, 2 or 3 — this alone can save 1–2 minutes per paper.
- SIGNS Before substituting, rewrite every negative term with brackets, e.g., treat (x−5) as a = x, b = −5 rather than risking a sign slip midway.
- SMART CHOICE When a question gives you a+b and ab (or a−b and ab), always check if it's secretly asking for a²+b², a³+b³, or a³−b³ — these can be solved WITHOUT finding a and b individually.
- SPECIAL CASE Whenever three terms in a question add to zero, immediately think of a³+b³+c³ = 3abc — this is a favourite HOTS and 3-4 mark question type.
- VERIFY If time permits, substitute small numbers (like a=1, b=2) into both sides of your final factorised/expanded answer to quickly self-check.
- PRESENTATION Box or underline your final answer — many State Board evaluators specifically look for a clearly marked final result.
✍️ Practice Question Bank
Attempt each question on paper first, then tap "Show Answer" to check. Resisting the urge to peek early is the single best study habit you can build.
Easy — Direct Application
Medium — Apply & Factorise
HOTS — Higher Order Thinking
🌟 Key Takeaways
- An identity is true for ALL values of the variables, unlike an equation which is true only for specific values.
- The first three identities — (a+b)², (a−b)², (a+b)(a−b) — are the most exam-frequent and worth memorising until they're instant reflexes.
- Cube identities follow the 1-3-3-1 pattern from Pascal's Triangle, with alternating signs for the difference form.
- Factorised forms (a³±b³) always pair an opposite-sign binomial with a same-sign trinomial in the second bracket.
- The "a+b+c=0" special case is a guaranteed shortcut for a³+b³+c³ = 3abc — examiners love testing this.
- You rarely need to find individual variable values — most problems can be solved directly using sums, products, and the identities, saving precious exam time.
❓ Your Questions, Answered
You've Got This 💪
Ten identities, one logic. Every formula on this page comes from the same basic idea — multiplying out brackets carefully and combining like terms. There's no magic here, only patterns, and you've now seen exactly where each one comes from. The next time (a+b)² or a³−b³ shows up on your test paper, you won't see a scary jumble of letters — you'll see an old friend.
Keep practising. Keep questioning "why," not just "what." That's how real Maths confidence is built — one identity, one problem, one small win at a time.
0 Comments