MathsICSE Class 8 · Algebraic Expressions and Identities

Identities (a ± b)² and a² − b²: Expansion, Shortcuts and Problems (ICSE Class 8)

Updated 17 October 2026 · Fact-checked

$(a + b)^2 = a^2 + 2ab + b^2$, $(a - b)^2 = a^2 - 2ab + b^2$ and $a^2 - b^2 = (a - b)(a + b)$. They give quick arithmetic such as $103^2 = (100 + 3)^2$ and link sums, differences and products: $a^2 + b^2 = (a + b)^2 - 2ab$.

Understand Identities: Squares and Difference of Squares

An identity is true for every value of the variables. The CISCE curriculum lists $(a \pm b)^{2} = a^{2} \pm 2 ab + b^{2}$ and $a^{2} - b^{2} = (a - b) (a + b)$ .

Why they hold. $(a + b)^{2} = (a + b) (a + b) = a^{2} + ab + ba + b^{2} = a^{2} + 2 ab + b^{2}$ . Replacing $b$ by $- b$ gives the difference form. The middle term $2 ab$ is the most commonly forgotten.

Expanding with larger terms. Treat the first and last terms as wholes: $(2 a - 3 b)^{2} = (2 a)^{2} - 2 (2 a) (3 b) + (3 b)^{2} = 4 a^{2} - 12 ab + 9 b^{2}$ .

Arithmetic shortcuts. Choose a nearby round number: $10 3^{2} = (100 + 3)^{2} = 10609$ ; $99 8^{2} = (1000 - 2)^{2} = 996004$ ; $97 \times 103 = (100 - 3) (100 + 3) = 9991$ ; $8. 7^{2} - 1. 3^{2} = (8.7 - 1.3) (8.7 + 1.3) = 74$ .

Connecting sum, difference and product.

$a^{2} + b^{2} = (a + b)^{2} - 2 ab = (a - b)^{2} + 2 ab$ .
$(a + b)^{2} - (a - b)^{2} = 4 ab$ .
$(a + b)^{2} + (a - b)^{2} = 2 (a^{2} + b^{2})$ .

Reciprocals. If $x + \frac{1}{x} = k$ , then $x^{2} + \frac{1}{x ^{2}} = k^{2} - 2$ , because the middle term of the square is $2 x \cdot \frac{1}{x} = 2$ . Repeating this gives higher powers.

Common error. $(a + b)^{2} \neq = a^{2} + b^{2}$ .

Identities: Squares and Difference of Squares: formulas to remember

Square of a sum: $(a + b)^{2} = a^{2} + 2 ab + b^{2}$; Do not omit $2 ab$ .
Square of a difference: $(a - b)^{2} = a^{2} - 2 ab + b^{2}$; Middle term is negative.
Difference of squares: $a^{2} - b^{2} = (a - b) (a + b)$; For shortcuts and factorising.
Reciprocal pair: $x^{2} + \frac{1}{x ^{2}} = (x + \frac{1}{x})^{2} - 2$; Also $= (x - \frac{1}{x})^{2} + 2$ .

How to solve Identities: Squares and Difference of Squares questions

Match the pattern, substitute carefully, and choose a round number for arithmetic.

1Identify which identity matches (sum square, difference square, or difference of squares).
2Name $a$ and $b$ clearly, including coefficients.
3Substitute and expand, keeping the middle term and its sign.
4For arithmetic, split the number as round $\pm$ small.
5Verify with a small substitution if needed.

Identities: Squares and Difference of Squares 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.

Mental arithmetic

Suppose a shopkeeper wants to compute the cost of 98 items at Rs 102 each without a calculator.

Use an identity to find the total.

Try it first, then show the solutionSolution

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

Answer: Rs 9,996.

Memory hook: Same centre, equal gap: difference of squares.

Tricks to remember Identities: Squares and Difference of Squares

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

Nearby round number
Write a number as 100 plus or minus a small amount and use the identity.
Use it when: Squares and products of near-round numbers.
The forgotten 2ab
The square of a sum has three terms, not two.
Use it when: Every expansion.

Identities: Squares and Difference of Squares: solved examples

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

Example 1: Expansion · 1 mark · MCQBoard level

$(x + 5)^{2}$ is equal to:
(a) $x^{2} + 25$ (b) $x^{2} + 10 x + 25$ (c) $x^{2} + 5 x + 25$ (d) $x^{2} + 10 x + 10$

Try it first, then show the solutionSolution

$a = x$ , $b = 5$ : $x^{2} + 2 (x) (5) + 25$ .

Answer: (b)

Example 2: Square of a near-round number · 2 marks · Short answerBoard level

Evaluate $10 3^{2}$ using an identity.

Try it first, then show the solutionSolution

$10 3^{2} = (100 + 3)^{2} = 10 0^{2} + 2 \times 100 \times 3 + 3^{2}$ .
$= 10000 + 600 + 9 = 10609$ .

Answer: 10609

Example 3: Product by difference of squares · 3 marks · Short answerHarder board level

Evaluate $97 \times 103$ without multiplying directly.

Try it first, then show the solutionSolution

$97 = 100 - 3$ and $103 = 100 + 3$ .
$(100 - 3) (100 + 3) = 10 0^{2} - 3^{2} = 10000 - 9 = 9991$ .

Answer: 9991

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 $(a - b)^{2}$ . Name the numbers.

Try it first, then show the solutionSolution

$a^{2} + b^{2} = (a + b)^{2} - 2 ab = 81 - 40 = 41$ .
$(a - b)^{2} = a^{2} + b^{2} - 2 ab = 41 - 40 = 1$ .
Check: $a, b = 5, 4$ gives sum $9$ , product $20$ .

Answer: $a^{2} + b^{2} = 41$ ; $(a - b)^{2} = 1$ .

Example 5: Reciprocal identity · 4 marks · Long answerHOTS / competency

If $x + \frac{1}{x} = 5$ , find $x^{2} + \frac{1}{x ^{2}}$ , $(x - \frac{1}{x})^{2}$ and $x^{4} + \frac{1}{x ^{4}}$ .

Try it first, then show the solutionSolution

$x^{2} + \frac{1}{x ^{2}} = 5^{2} - 2 = 23$ .
$(x - \frac{1}{x})^{2} = 23 - 2 = 21$ .
$x^{4} + \frac{1}{x ^{4}} = (x^{2} + \frac{1}{x ^{2}})^{2} - 2 = 529 - 2 = 527$ .

Answer: $23$ ; $21$ ; $527$ .

Where students lose marks in Identities: Squares and Difference of Squares

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

Mistake: Writing (a + b)^2 = a^2 + b^2
The middle term 2ab is missing.
Wrong working
$(x + 5)^{2} = x^{2} + 25$ .
Right working
$x^{2} + 10 x + 25$ .
Do this: Write three terms every time.
Usually costs: Answer mark.
Mistake: Wrong sign of 2ab in the difference square
The middle term is negative.
Wrong working
$(x - 4)^{2} = x^{2} + 8 x + 16$ .
Right working
$x^{2} - 8 x + 16$ .
Do this: Sign of the middle term follows the sign inside the bracket.
Usually costs: Answer mark.

Exam tips

Write which identity you use as a line of working.
For arithmetic shortcuts, show the split (100 plus or minus a gap).
Remember the identity gives an equality only, not a way to remove the middle term.

Practice questions on Identities: Squares and Difference of Squares

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
$(2 a - 3 b)^{2}$ equals:
(a) $4 a^{2} - 9 b^{2}$ (b) $4 a^{2} + 9 b^{2}$ (c) $4 a^{2} - 12 ab + 9 b^{2}$ (d) $2 a^{2} - 6 ab + 3 b^{2}$
Hint
Square the first, minus twice the product, square the last.
Answer
(c)
Q2 · 2 marks · Short answerBoard level
Evaluate $99 8^{2}$ using an identity.
Hint
998 = 1000 - 2.
Answer
$(1000 - 2)^{2} = 1000000 - 4000 + 4 = 996004$ .
Q3 · 3 marks · Short answerHarder board level
Evaluate $8. 7^{2} - 1. 3^{2}$ .
Hint
Difference of squares.
Answer
$(8.7 - 1.3) (8.7 + 1.3) = 7.4 \times 10 = 74$ .
Q4 · 3 marks · Short answerHarder board level
If $x - y = 4$ and $x y = 12$ , find $x^{2} + y^{2}$ and $(x + y)^{2}$ .
Hint
Use (x - y)^2 + 2xy.
Answer
$x^{2} + y^{2} = 16 + 24 = 40$ . $(x + y)^{2} = 40 + 24 = 64$ .
Q5 · 4 marks · Long answerHOTS / competency
Prove that $(a + b)^{2} - (a - b)^{2} = 4 ab$ . Use it to evaluate $(7.5 + 2.4)^{2} - (7.5 - 2.4)^{2}$ .
Hint
Expand both squares or use the identity directly.
Answer
$(a^{2} + 2 ab + b^{2}) - (a^{2} - 2 ab + b^{2}) = 4 ab$ . With $a = 7.5, b = 2.4$ : $4 \times 7.5 \times 2.4 = 72$ . Check: $9. 9^{2} - 5. 1^{2} = 98.01 - 26.01 = 72$ .

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.(a - b)^2 = ?Show answer
a^2 - 2ab + b^2.
2.a^2 - b^2 = ?Show answer
(a - b)(a + b).
3.a^2 + b^2 in terms of a + b and ab?Show answer
(a + b)^2 - 2ab.
4.If x + 1/x = k, x^2 + 1/x^2 = ?Show answer
k^2 - 2.

Identities: Squares and Difference of Squares: frequently asked questions

Is (a + b)^2 the same as a^2 + b^2?

No; there is an extra 2ab.

Can I use these identities with fractions or decimals?

Yes; they hold for all numbers.