Page 148 - ICSE Math 8

13 Algebraic Identities

Key Concepts

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• Standard Identities: (a + b) = a + 2ab + b 2 • Special Products
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(a – b) = a – 2ab + b 2
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a – b = (a – b)(a + b)
An algebraic identity is a statement of equality between two algebraic expressions that is satisfied for all the
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values of the variables. For example, (x – 3)(x + 2) ∫ x – x – 6 is satisfied for all values of x. The sign ‘∫’ is
used to distinguish an identity from an equation.
Standard Identities

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Identity 1: (a + b) = a + 2ab + b 2
In other words,
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(Sum of the two terms) = (First term) + 2 ¥ (First term) ¥ (Second term) + (Second term) 2
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Proof: (a + b) = (a + b)(a + b)
= a(a + b) + b(a + b) (Distributive property of multiplication over addition)
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= a + ab + ab + b 2
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= a + 2ab + b (Commutative property ab = ba)
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\ (a + b) = a + 2ab + b 2
Verification
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Let us verify the identity (a + b) = a + 2ab + b by performing the following activity.
Step 1: Take the black sheet of paper and cut a square of side equal to a + b, where a = 3, b = 5. Paste it on
the white sheet.
Step 2: Now cut two squares of sides a (3 cm) and b (5 cm) from another blue and green sheets.
Step 3: Cut another two rectangles of dimensions 3 cm ¥ 5 cm each from a pink sheet.
Step 4: Now paste one side of each square or rectangle cut on the black sheet as shown in the given figure.

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(a + b) = 8 × 8 = 64 cm 2
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a + ab + ab + b 2
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= a + 2ab + b 2
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= (3) + 2(3)(5) + (5) 2
= 9 + 30 + 25 = 64 cm 2
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\ (a + b) = a + 2ab + b 2
Hence verified.
Now repeat for different
values of a and b.

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