Algebraic Expressions and Identities: Exercise 9.5 - Class 8 Maths

Understanding the Four Standard Algebraic Identities

Algebraic identities are fundamental equations that are true for every possible value of the variables involved. They are incredibly useful for expanding expressions, factoring, and simplifying calculations, especially when dealing with squares and products of binomials. In Class 8, you will primarily work with four key identities. Let's look at them:

1. Identity I: (a + b)² = a² + 2ab + b²

This identity states that the square of the sum of two terms is equal to the square of the first term, plus twice the product of the two terms, plus the square of the second term.
Example: (x + 3)² = x² + 2(x)(3) + 3² = x² + 6x + 9

1. Identity II: (a - b)² = a² - 2ab + b²

This identity states that the square of the difference of two terms is equal to the square of the first term, minus twice the product of the two terms, plus the square of the second term.
Example: (2y - 5)² = (2y)² - 2(2y)(5) + 5² = 4y² - 20y + 25

1. Identity III: (a + b)(a - b) = a² - b²

This identity states that the product of the sum and difference of two terms is equal to the square of the first term minus the square of the second term. This is often called the 'Difference of Squares' identity.
Example: (4p + 7)(4p - 7) = (4p)² - 7² = 16p² - 49

1. Identity IV: (x + a)(x + b) = x² + (a + b)x + ab

This identity is used when two binomials have one common term ('x') and different constant terms ('a' and 'b'). It expands to the square of the common term, plus the sum of the constant terms multiplied by the common term, plus the product of the constant terms.
Example: (z + 2)(z + 5) = z² + (2 + 5)z + (2)(5) = z² + 7z + 10

Understanding these identities thoroughly is the first step to mastering Exercise 9.5. They provide a systematic way to expand and multiply expressions without having to do lengthy term-by-term multiplications.

Step-by-Step Process for Applying Identities

1. Step 1: Observe the Expression's Form — Look closely at the algebraic expression you need to simplify or expand. Does it resemble (a+b)², (a-b)², (a+b)(a-b), or (x+a)(x+b)? Identify the number of terms, the operations (addition/subtraction), and if any terms are squared or multiplied in a specific pattern. For example, if you see 'something squared', think of Identity I or II. If you see 'two binomials multiplied', think of Identity III or IV.
2. Step 2: Choose the Correct Identity — Based on your observation in Step 1, select the most appropriate algebraic identity. For instance, if you have (5x + 2y)², it clearly matches (a+b)². If you have (98)², you can rewrite it as (100 - 2)² to use (a-b)².
3. Step 3: Identify 'a', 'b', and 'x' (if applicable) — Once you've chosen an identity, carefully identify what 'a' and 'b' (and 'x' in Identity IV) represent in your specific expression. Be careful with signs. For example, in (3m - 4n)², 'a' would be 3m and 'b' would be 4n (not -4n, as the identity itself has the minus sign).
4. Step 4: Substitute and Expand — Substitute the identified 'a', 'b', and 'x' values into the chosen identity's formula. Perform the necessary multiplications and squaring operations. Remember to use parentheses for terms like (3m)² to ensure the entire term is squared, not just 'm'.
5. Step 5: Simplify the Result — After substituting and expanding, combine any like terms to get the final simplified algebraic expression. This is your answer. Always double-check your calculations, especially signs.

Worked Examples Using Algebraic Identities

• Example 1: Expand (3x + 4y)² Step 1: Identify the form. This is in the form (a + b)². Step 2: Identify 'a' and 'b'. Here, a = 3x and b = 4y. Step 3: Apply Identity I: (a + b)² = a² + 2ab + b². Step 4: Substitute 'a' and 'b' into the identity: (3x + 4y)² = (3x)² + 2(3x)(4y) + (4y)² Step 5: Simplify: = 9x² + 24xy + 16y² Final Answer: 9x² + 24xy + 16y²
• Example 2: Find the value of (99)² using an identity. Step 1: Identify the form. We can rewrite 99 as (100 - 1), which is in the form (a - b)². Step 2: Identify 'a' and 'b'. Here, a = 100 and b = 1. Step 3: Apply Identity II: (a - b)² = a² - 2ab + b². Step 4: Substitute 'a' and 'b' into the identity: (100 - 1)² = (100)² - 2(100)(1) + (1)² Step 5: Simplify: = 10000 - 200 + 1 = 9801 Final Answer: 9801
• Example 3: Evaluate (5p + 3q)(5p - 3q) Step 1: Identify the form. This is in the form (a + b)(a - b). Step 2: Identify 'a' and 'b'. Here, a = 5p and b = 3q. Step 3: Apply Identity III: (a + b)(a - b) = a² - b². Step 4: Substitute 'a' and 'b' into the identity: (5p + 3q)(5p - 3q) = (5p)² - (3q)² Step 5: Simplify: = 25p² - 9q² Final Answer: 25p² - 9q²
• Example 4: Expand (y + 6)(y + 8) Step 1: Identify the form. This is in the form (x + a)(x + b). Step 2: Identify 'x', 'a', and 'b'. Here, x = y, a = 6, and b = 8. Step 3: Apply Identity IV: (x + a)(x + b) = x² + (a + b)x + ab. Step 4: Substitute 'x', 'a', and 'b' into the identity: (y + 6)(y + 8) = y² + (6 + 8)y + (6)(8) Step 5: Simplify: = y² + 14y + 48 Final Answer: y² + 14y + 48

Exam Tip: Avoiding Common Mistakes

When applying algebraic identities, students often make a few common errors. Being aware of these can help you avoid losing marks:

1. Sign Errors: This is perhaps the most frequent mistake. Remember that in (a - b)², the middle term is -2ab, not +2ab. Also, in (a+b)(a-b) = a² - b², the minus sign between a² and b² is crucial.
2. Incomplete Squaring: In (a + b)² = a² + 2ab + b², some students might forget to square both terms 'a' and 'b' completely, or might only square part of a term (e.g., in (3x)², writing 3x² instead of 9x²).
3. Confusing Identities: Make sure you correctly identify which identity to use. For example, don't confuse (a+b)² with (x+a)(x+b). Look for the specific pattern (e.g., 'entire expression squared' vs. 'product of two different binomials').
4. Careless Calculation: After applying the identity, always simplify carefully, combining like terms and performing arithmetic operations accurately. Pay attention to multiplication of coefficients and variables.

Practice Questions with Solutions

• Q: Use a suitable identity to find the product: (x + 5)(x + 2) A: Step 1: Identify the form. This is in the form (x + a)(x + b). Step 2: Identify x, a, and b. Here, x = x, a = 5, b = 2. Step 3: Apply Identity IV: (x + a)(x + b) = x² + (a + b)x + ab. Step 4: Substitute the values: x² + (5 + 2)x + (5)(2) Step 5: Simplify: x² + 7x + 10 Final answer: x² + 7x + 10
• Q: Find the product: (2a - 7)(2a - 7) A: Step 1: Identify the form. This can be written as (2a - 7)², which is in the form (a - b)². Step 2: Identify 'a' and 'b'. Here, a = 2a and b = 7. Step 3: Apply Identity II: (a - b)² = a² - 2ab + b². Step 4: Substitute the values: (2a)² - 2(2a)(7) + (7)² Step 5: Simplify: 4a² - 28a + 49 Final answer: 4a² - 28a + 49
• Q: Use a suitable identity to evaluate (50 - 3)(50 + 3). A: Step 1: Identify the form. This is in the form (a - b)(a + b). Step 2: Identify 'a' and 'b'. Here, a = 50 and b = 3. Step 3: Apply Identity III: (a + b)(a - b) = a² - b². Step 4: Substitute the values: (50)² - (3)² Step 5: Simplify: 2500 - 9 = 2491 Final answer: 2491
• Q: Expand (1/2 p + 3/4 q)² using a suitable identity. A: Step 1: Identify the form. This is in the form (a + b)². Step 2: Identify 'a' and 'b'. Here, a = 1/2 p and b = 3/4 q. Step 3: Apply Identity I: (a + b)² = a² + 2ab + b². Step 4: Substitute the values: (1/2 p)² + 2(1/2 p)(3/4 q) + (3/4 q)² Step 5: Simplify: 1/4 p² + (3/4)pq + 9/16 q² Final answer: 1/4 p² + 3/4 pq + 9/16 q²

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