01 · Explore
Understanding Algebraic Identities
An algebraic identity is a mathematical equation that holds true for every possible value of the variables involved, distinguishing it from a standard equation.
In earlier grades, you worked with equations like x² - 1 = 24. This is only true when x = 5 or -5. However, an identity like (x + y)² = x² + 2xy + y² is true regardless of what numbers you substitute for x and y. Identities serve as powerful shortcuts, allowing us to expand products or factorise expressions without long multiplication.
Consider the pattern of three consecutive square numbers. If we let the middle number be 'n', the three squares are (n - 1)², n², and (n + 1)². Adding the smallest and largest squares and subtracting twice the middle square gives: (n - 1)² + (n + 1)² - 2n² = (n² - 2n + 1) + (n² + 2n + 1) - 2n² = 2n² + 2 - 2n² = 2. This result is always 2, regardless of the value of n.
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Question
What is the primary difference between an algebraic equation and an algebraic identity?
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Sign inNew here? Sign up freeNCERT reference: chapter PDF pages 1, 3, 4, 6.
02 · Explore
The Square of a Binomial Sum
The identity (a + b)² = a² + 2ab + b² allows us to find the square of the sum of two terms efficiently.
We can visualise this identity by constructing a square with side length (a + b). If we partition this square, we get one square of area a², one square of area b², and two rectangles each with area ab. Summing these areas gives the total area (a + b)², which equals a² + 2ab + b².
This identity applies to all real numbers, including negative and rational numbers. For example, if a = -2 and b = -3, then (a + b)² = (-5)² = 25. Using the identity: (-2)² + 2(-2)(-3) + (-3)² = 4 + 12 + 9 = 25. The results match perfectly.
Expanding a Binomial Expression
(5x + 2y)²
Identify a = 5x and b = 2y. Apply the identity a² + 2ab + b²: (5x)² + 2(5x)(2y) + (2y)² = 25x² + 20xy + 4y².
Numerical Calculation using Identities
43² = (40 + 3)²
Let a = 40 and b = 3. Using the identity: 40² + 2(40)(3) + 3² = 1600 + 240 + 9 = 1849.
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Question
Is (a + b)² always equal to a² + b²?
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03 · Explore
The Square of a Binomial Difference
By replacing b with -b in the previous identity, we derive the identity for the square of a difference: (a - b)² = a² - 2ab + b².
Geometrically, to obtain (a - b)², we start with a square of side 'a' (area a²). We subtract a rectangle of area ab and another rectangle of area b(a - b). This process removes the area of the rectangles from the big square to leave the small square of side (a - b).
This identity is particularly useful for squaring numbers just below a base of 10 or 100. For instance, 29² can be viewed as (30 - 1)².
Calculating 29²
(30 - 1)²
Apply (a - b)² = a² - 2ab + b² where a = 30 and b = 1. Calculation: 30² - 2(30)(1) + 1² = 900 - 60 + 1 = 841.
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Question
Expand (2x - 3y)² using the identity.
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04 · Explore
The Square of a Trinomial
The identity for the square of a sum of three numbers is (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca.
To derive this, we can treat (b + c) as a single term 'd'. Then (a + d)² = a² + 2ad + d². Substituting (b + c) back for 'd' and expanding (b + c)² gives us the full trinomial expansion.
This identity is useful for expanding expressions with three terms and for finding the sum of squares if the sum of the numbers and the sum of their pairwise products are known.
Squaring 119
(100 + 10 + 9)²
Using a=100, b=10, c=9: 100² + 10² + 9² + 2(100)(10) + 2(100)(9) + 2(10)(9) = 10000 + 100 + 81 + 2000 + 1800 + 180 = 14161.
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Question
If a + b + c = 10 and ab + bc + ca = 31, find a² + b² + c².
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05 · Explore
Product of Sum and Difference
The identity a² - b² = (a + b)(a - b) is a fundamental rule for both expansion and factorisation.
This identity tells us that the difference of two squares is equal to the product of the sum and the difference of the two terms. It can also be rewritten as a² = (a + b)(a - b) + b².
In 750 CE, the mathematician Śhrīdharāchārya proposed this as a method to compute squares quickly. For example, to find 55², we can use (55 + 5)(55 - 5) + 5² = 60 × 50 + 25 = 3000 + 25 = 3025.
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Question
Factorise 49x² - 16y².
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Sign inNew here? Sign up freeNCERT reference: chapter PDF page 10.