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medium difficulty5 min

Square Identities

Square identities are standard expansions for (a+b)2, (a-b)2, and (a+b)(a-b) used to simplify products quickly.

In one line: Square of binomial gives first square, double product, last square.

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Understand it

The three main square identities are (a+b)2=a2+2ab+b2, (a-b)2=a2-2ab+b2, and (a+b)(a-b)=a2-b2. They save time because the middle term and signs can be written by pattern recognition.

Example

(x+5)2 = x2+10x+25 and (3p-2)2 = 9p2-12p+4.

Formula and rule

(a+b)2=a2+2ab+b2; (a-b)2=a2-2ab+b2; (a+b)(a-b)=a2-b2.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Expand (2x-3)2.

  1. 1Here a=2x and b=3.
  2. 2So (2x)2-2(2x)(3)+32 = 4x2-12x+9.

Method you can reuse

Identify a and b, choose the correct identity by checking the sign and product form, substitute carefully, then simplify powers and coefficients.

Visual tip

A square split into side lengths a and b can show a2, b2, and two ab rectangles for (a+b)2.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Why is (x-4)2 not equal to x2-16?

Think first, then show answer
Because (x-4)2 uses the square identity and expands to x2-8x+16. The middle term -8x cannot be skipped.

Common wrong answer

(2x-3)2=4x2-9.

Why it loses marks: It applies difference of squares to a squared binomial.

Don't mix this up

Students often write (a-b)2 as a2-b2, forgetting the middle term -2ab.

Exam tip

For a square of a binomial, always write three terms; for sum multiplied by difference, write two terms.

Square Identities diagram

Open full screen to read every label clearly.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Questions often ask expansion, simplification, or quick calculation such as 1032 using (100+3)2.
  1. 1

    Enough for 1 mark

    Square identities are standard expansions for (a+b)2, (a-b)2, and (a+b)(a-b) used to simplify products quickly.

  2. 2

    Add for 2 marks

    Example: (x+5)2 = x2+10x+25 and (3p-2)2 = 9p2-12p+4.

  3. 3

    Add for 3 marks

    The three main square identities are (a+b)2=a2+2ab+b2, (a-b)2=a2-2ab+b2, and (a+b)(a-b)=a2-b2.

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