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Square Identities

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

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Student-friendly explanation

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

How to write this in exams

  1. 1

    Start with the exact idea

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

  2. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

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

  4. 4

    Avoid this incomplete answer

    (2x-3)^2=4x^2-9 is wrong because it applies difference of squares to a squared binomial.

Definition

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

Example

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

Rule to remember

(a+b)^2=a^2+2ab+b^2; (a-b)^2=a^2-2ab+b^2; (a+b)(a-b)=a^2-b^2.

Memory hook

Square of binomial gives first square, double product, last square.

Examples and method

Worked example

Expand (2x-3)^2. Here a=2x and b=3. So (2x)^2-2(2x)(3)+3^2 = 4x^2-12x+9.

Method to apply

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

Diagram support

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

How CBSE asks it

Questions often ask expansion, simplification, or quick calculation such as 103^2 using (100+3)^2.

Avoid common mistakes

Common confusion

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

Common wrong answer

(2x-3)^2=4x^2-9 is wrong because it applies difference of squares to a squared binomial.

Exam tip

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

Quick check

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

Because (x-4)^2 uses the square identity and expands to x^2-8x+16. The middle term -8x cannot be skipped.

Study the square identities diagram carefully

Use the labelled diagram to keep square identities clear in short answers and revision.

What this diagram makes clear

This diagram keeps the labels and direction of square identities in the right order.

Where this helps in exams

Use this for labelled diagram work and short exam answers on square identities.

Revision cue

Revise square identities through the labels before writing the answer.

Answer writing and exam use

1-mark answer

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

2-mark answer

Square identities are standard expansions for (a+b)^2, (a-b)^2, and (a+b)(a-b) used to simplify products quickly. (a+b)^2=a^2+2ab+b^2; (a-b)^2=a^2-2ab+b^2; (a+b)(a-b)=a^2-b^2. (x+5)^2 = x^2+10x+25 and (3p-2)^2 = 9p^2-12p+4.

3-mark answer

The three main square identities are (a+b)^2=a^2+2ab+b^2, (a-b)^2=a^2-2ab+b^2, and (a+b)(a-b)=a^2-b^2. They save time because the middle term and signs can be written by pattern recognition. (a+b)^2=a^2+2ab+b^2; (a-b)^2=a^2-2ab+b^2; (a+b)(a-b)=a^2-b^2. Expand (2x-3)^2. Here a=2x and b=3. So (2x)^2-2(2x)(3)+3^2 = 4x^2-12x+9. Questions often ask expansion, simplification, or quick calculation such as 103^2 using (100+3)^2. (2x-3)^2=4x^2-9 is wrong because it applies difference of squares to a squared binomial.
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