MathsICSE Class 9 · Expansions

Square Identities: (a ± b)², (x + a)(x + b) and (a + b + c)² with Solved Questions (ICSE Class 9)

Updated 17 October 2026 · Fact-checked

The identities are (a ± b)² = a² ± 2ab + b², (x + a)(x + b) = x² + (a + b)x + ab and (a + b + c)² = a² + b² + c² + 2(ab + bc + ca). They expand brackets quickly, evaluate numbers such as 103² and 997 × 1003, and connect sums and products: a² + b² = (a + b)² − 2ab.

Notice the two orange ab rectangles: they are the middle term 2ab that students often forget when expanding (a + b)².

Illustration drawn for QuestionBuddy.

Understand Square of a Binomial, Product (x + a)(x + b) and Square of a Trinomial

Binomial squares.
$ $(a + b)^{2} = a^{2} + 2 ab + b^{2}, (a - b)^{2} = a^{2} - 2 ab + b^{2} .$ $
Each term in the middle is twice the product of the two terms. So $(3 x - 2 y)^{2} = 9 x^{2} - 12 x y + 4 y^{2}$ .

Product of two binomials with a common term.
$ $(x + a) (x + b) = x^{2} + (a + b) x + ab .$ $
The coefficient of $x$ is the sum of the two numbers; the constant is their product. This identity is also the idea behind factorising quadratics.

Square of a trinomial.
$ $(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 ab + 2 b c + 2 c a .$ $
Signs go with the terms: $(2 x + 3 y - z)^{2}$ has cross terms $2 (2 x) (3 y) = 12 x y$ , $2 (3 y) (- z) = - 6 y z$ and $2 (- z) (2 x) = - 4 z x$ .

Evaluating numbers. Write the number near a round number: $10 3^{2} = (100 + 3)^{2} = 10609$ ; $102 \times 98 = (100 + 2) (100 - 2) = 10 0^{2} - 2^{2} = 9996$ ; $997 \times 1003 = 100 0^{2} - 3^{2} = 999991$ .

Sums, products and squares. From $(a + b)^{2}$ and $(a - b)^{2}$ :
$ $a^{2} + b^{2} = (a + b)^{2} - 2 ab = (a - b)^{2} + 2 ab, (a + b)^{2} - (a - b)^{2} = 4 ab .$ $
For three numbers, $a^{2} + b^{2} + c^{2} = (a + b + c)^{2} - 2 (ab + b c + c a)$ . These let you find a combination (like $a^{2} + b^{2}$ ) without finding $a$ and $b$ .

Check. Test an expansion with a small number such as $x = 1, y = 1$ ; both sides must agree. This catches sign slips quickly.

Level. Class IX is a school-exam class. Items at the 'advanced' level combine two identities; the 'hots' items add a consistency check and are at the top of the school range.

Square of a Binomial, Product (x + a)(x + b) and Square of a Trinomial: formulas to remember

Binomial square: $(a \pm b)^{2} = a^{2} \pm 2 ab + b^{2}$; Middle term is twice the product.
Common-term product: $(x + a) (x + b) = x^{2} + (a + b) x + ab$; Sum and product.
Trinomial square: $(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 (ab + b c + c a)$; Carry the signs.
Sum of squares: $a^{2} + b^{2} = (a + b)^{2} - 2 ab$; Or (a − b)² + 2ab.
Difference of squares: $(a + b)^{2} - (a - b)^{2} = 4 ab$; Useful in simplifications.

How to solve Square of a Binomial, Product (x + a)(x + b) and Square of a Trinomial questions

Choose the identity, label a, b (and c), then expand with signs.

1Decide which identity matches the expression.
2Label the terms $a$ , $b$ (with their signs).
3Write each term of the expansion separately, then combine like terms.
4For numbers, split into a round number plus or minus a small one.
5For sum-and-product questions, express the target using $(a \pm b)^{2}$ and the given values.
6Test the final result with $x = 1$ (and $y = 1$ ).

Square of a Binomial, Product (x + a)(x + b) and Square of a Trinomial in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Square plot with a bigger side

Suppose a square tile floor has side $(2 x + 5)$ m and is extended so that each side grows by 2 m.

By how much does the area increase, as an expression in $x$ ? Find it when $x = 4$ .

Try it first, then show the solutionSolution

New area $-$ old area $= (2 x + 7)^{2} - (2 x + 5)^{2} = (4 x^{2} + 28 x + 49) - (4 x^{2} + 20 x + 25) = 8 x + 24$ .
At $x = 4$ : $8 (4) + 24 = 56$ m² (check: $1 5^{2} - 1 3^{2} = 225 - 169 = 56$ ).

Answer: $8 x + 24$ m²; 56 m² when $x = 4$ .

Memory hook: Expand both squares, then subtract.

Mental maths with a shop price

Imagine a shopkeeper pricing 102 items at ₹98 each in his head.

How can the product be found without long multiplication?

Try it first, then show the solutionSolution

$102 \times 98 = (100 + 2) (100 - 2) = 10 0^{2} - 2^{2} = 10000 - 4 = 9996$ .

Answer: ₹9,996.

Memory hook: Pair numbers around a round number.

Tricks to remember Square of a Binomial, Product (x + a)(x + b) and Square of a Trinomial

Short memory aids. Say each one aloud once, then use it on a question the same day.

Twice the product
The middle term is always 2ab, never ab.
Use it when: Binomial expansions.
Sum and product
(x + a)(x + b): sum goes with x, product is the constant.
Use it when: Expansion and factorisation.
Three squares, three pairs
A trinomial square has 3 squares and 3 cross terms with a factor of 2.
Use it when: Trinomial expansions.

Square of a Binomial, Product (x + a)(x + b) and Square of a Trinomial: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Expanding a binomial square · 1 mark · MCQBoard level

$(3 x - 2 y)^{2}$ equals:
(a) $9 x^{2} - 4 y^{2}$
(b) $9 x^{2} - 6 x y + 4 y^{2}$
(c) $9 x^{2} - 12 x y + 4 y^{2}$
(d) $9 x^{2} + 12 x y + 4 y^{2}$

Try it first, then show the solutionSolution

$a = 3 x$ , $b = 2 y$ : $a^{2} - 2 ab + b^{2} = 9 x^{2} - 12 x y + 4 y^{2}$ .

Answer: (c)

Example 2: Evaluating with identities · 3 marks · Short answerBoard level

Using identities, evaluate (i) $10 3^{2}$ (ii) $9 8^{2}$ (iii) $102 \times 98$ .

Try it first, then show the solutionSolution

(i) $(100 + 3)^{2} = 10000 + 600 + 9 = 10609$ .
(ii) $(100 - 2)^{2} = 10000 - 400 + 4 = 9604$ .
(iii) $(100 + 2) (100 - 2) = 10000 - 4 = 9996$ .

Answer: 10609; 9604; 9996

Example 3: Trinomial square · 4 marks · Short answerHarder board level

Expand $(2 x + 3 y - z)^{2}$ and find its value when $x = 1$ , $y = 1$ , $z = 2$ .

Try it first, then show the solutionSolution

$a = 2 x, b = 3 y, c = - z$ .
$= 4 x^{2} + 9 y^{2} + z^{2} + 12 x y - 6 y z - 4 z x$ .
At $(1, 1, 2)$ : $4 + 9 + 4 + 12 - 12 - 8 = 9$ .
Check: $(2 + 3 - 2)^{2} = 9$ . ✓

Answer: $4 x^{2} + 9 y^{2} + z^{2} + 12 x y - 6 y z - 4 z x$ ; value $9$ .

Example 4: From sum and product · 4 marks · Short answerHarder board level

If $a + b = 9$ and $ab = 20$ , find $a^{2} + b^{2}$ and $(a - b)^{2}$ , and hence $a$ and $b$ .

Try it first, then show the solutionSolution

$a^{2} + b^{2} = (a + b)^{2} - 2 ab = 81 - 40 = 41$ .
$(a - b)^{2} = (a + b)^{2} - 4 ab = 81 - 80 = 1$ , so $a - b = \pm 1$ .
With $a + b = 9$ : $a = 5, b = 4$ or $a = 4, b = 5$ .
Check: $25 + 16 = 41$ . ✓

Answer: $41$ ; $1$ ; the numbers are $4$ and $5$ .

Example 5: Three numbers · 5 marks · Long answerHOTS / competency

If $a + b + c = 10$ and $ab + b c + c a = 31$ , find $a^{2} + b^{2} + c^{2}$ and $a^{2} + b^{2} + c^{2} - ab - b c - c a$ . Find one set of positive integers satisfying both conditions.

Try it first, then show the solutionSolution

$a^{2} + b^{2} + c^{2} = 1 0^{2} - 2 (31) = 100 - 62 = 38$ .
$a^{2} + b^{2} + c^{2} - ab - b c - c a = 38 - 31 = 7$ .
Try integers with sum 10 and squares summing to 38: $25 + 9 + 4 = 38$ gives $5, 3, 2$ .
Check: $5 + 3 + 2 = 10$ ; $15 + 6 + 10 = 31$ . ✓

Answer: $38$ and $7$ ; for example $5, 3, 2$ .

Where students lose marks in Square of a Binomial, Product (x + a)(x + b) and Square of a Trinomial

The slips examiners see most often, and what to write instead.

Mistake: Missing the 2 in the middle term
The cross term is 2ab.
Wrong working
$(a + b)^{2} = a^{2} + ab + b^{2}$ .
Right working
$(a + b)^{2} = a^{2} + 2 ab + b^{2}$ .
Do this: Test with a = b = 1.
Usually costs: Whole expansion.
Mistake: Sign of the middle term in (a − b)²
The square of b is positive.
Wrong working
$(x - 3)^{2} = x^{2} - 9$ .
Right working
$(x - 3)^{2} = x^{2} - 6 x + 9$ .
Do this: Write a, b with signs first.
Usually costs: Whole expansion.
Mistake: Dropping the 2 in a trinomial
Every cross term has factor 2.
Wrong working
$(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + ab + b c + c a$ .
Right working
$a^{2} + b^{2} + c^{2} + 2 ab + 2 b c + 2 c a$ .
Do this: List three squares, then three doubled products.
Usually costs: Whole expansion.

Exam tips

Class IX is examined in school papers; sum-and-product items like these are common 3 to 4 mark questions.
Always show the identity before substituting; the method mark depends on it.
When asked for a combination (a² + b²), do not solve for a and b.
Check by substituting a small value for each variable.

Practice questions on Square of a Binomial, Product (x + a)(x + b) and Square of a Trinomial

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
The coefficient of $x$ in the expansion of $(x + 5) (x + 7)$ is:
(a) 2
(b) 12
(c) 35
(d) 5
Hint
Sum of 5 and 7.
Answer
(b) 12
Q2 · 3 marks · Short answerHarder board level
Evaluate using identities: (i) $997 \times 1003$ (ii) $(2.04)^{2}$ .
Hint
$(1000 - 3) (1000 + 3)$ and $(2 + 0.04)^{2}$ .
Answer
(i) $1000000 - 9 = 999991$ . (ii) $4 + 0.16 + 0.0016 = 4.1616$ .
Q3 · 4 marks · Short answerHarder board level
If $x - y = 4$ and $x y = 21$ , find $x^{2} + y^{2}$ and the possible values of $x + y$ .
Hint
$x^{2} + y^{2} = (x - y)^{2} + 2 x y$ .
Answer
$x^{2} + y^{2} = 16 + 42 = 58$ . $(x + y)^{2} = 16 + 84 = 100$ , so $x + y = \pm 10$ .
Q4 · 1 mark · Assertion-reasonHarder board level
Assertion (A): $(a - b)^{2} = (b - a)^{2}$ for all numbers $a$ and $b$ .
Reason (R): The squares of two opposite numbers are equal.
(a) Both (A) and (R) are true and (R) explains (A).
(b) Both are true but (R) does not explain (A).
(c) (A) is true, (R) is false.
(d) (A) is false, (R) is true.
Hint
$b - a = - (a - b)$ .
Answer
(a)
Q5 · 4 marks · Case-basedHOTS / competency
Case study: A square floor has side $(2 x + 5)$ metres.
(a) Write its area as an expression in $x$ .
(b) Find the area when $x = 4$ .
(c) The side is increased by 2 m. Find the increase in area in terms of $x$ , and its value at $x = 4$ .
Hint
Expand (2x + 7)² − (2x + 5)².
Answer
(a) $4 x^{2} + 20 x + 25$ (b) $169$ m² (c) $8 x + 24$ ; at $x = 4$ it is $56$ m² (since $225 - 169 = 56$ ).

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Expand (a + b)².Show answer
a² + 2ab + b².
2.Expand (x + a)(x + b).Show answer
x² + (a + b)x + ab.
3.a² + b² in terms of a + b and ab?Show answer
(a + b)² − 2ab.
4.Cross terms in (a + b + c)²?Show answer
2ab + 2bc + 2ca.

Square of a Binomial, Product (x + a)(x + b) and Square of a Trinomial: frequently asked questions

Why is the middle term 2ab?

The product (a + b)(a + b) gives ab twice, once from each bracket.

Can I use (a − b)² for numbers like 98²?

Yes: 98 = 100 − 2.

Do these identities appear in Class IX only?

They are used again in factorisation and in later classes.