What is (a+b) Whole Square?

S3-SA1-0041

Grade Level:

Class 6

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The expression "(a+b) Whole Square" means multiplying the sum of two numbers, 'a' and 'b', by itself. It's written as (a+b)^2. The result is a special algebraic identity: a^2 + 2ab + b^2.

Simple Example

Quick Example

Imagine you have two friends, Rohan and Priya. Rohan scored 'a' runs and Priya scored 'b' runs in a cricket match. If you want to find the square of their total runs, you'd calculate (a+b)^2. This would be the same as (Rohan's runs)^2 + 2 * (Rohan's runs) * (Priya's runs) + (Priya's runs)^2.

Worked Example

Step-by-Step

Let's find the value of (5+3)^2 using the identity.

Step 1: Identify 'a' and 'b'. Here, a = 5 and b = 3.
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Step 2: Recall the identity: (a+b)^2 = a^2 + 2ab + b^2.
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Step 3: Substitute the values of 'a' and 'b' into the identity: (5)^2 + 2 * (5) * (3) + (3)^2.
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Step 4: Calculate the squares: 5^2 = 25 and 3^2 = 9.
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Step 5: Calculate the middle term: 2 * 5 * 3 = 30.
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Step 6: Add all the terms: 25 + 30 + 9.
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Step 7: Perform the addition: 25 + 30 + 9 = 64.
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Step 8: Alternatively, (5+3)^2 = (8)^2 = 64. Both methods give the same answer.

Answer: (5+3)^2 = 64.

Why It Matters

Understanding (a+b)^2 is super important for solving complex problems in science and technology. Engineers use it when designing buildings or bridges, and computer scientists use it in creating efficient algorithms. Even data scientists use similar concepts to analyze patterns in large datasets, helping companies make smart decisions.

Common Mistakes

MISTAKE: Students often write (a+b)^2 as a^2 + b^2. | CORRECTION: Remember the middle term! The correct expansion is a^2 + 2ab + b^2. Don't forget to multiply 'a' and 'b' by 2.

MISTAKE: Incorrectly calculating 2ab, sometimes forgetting to multiply by 2 or making a calculation error. | CORRECTION: Always calculate 'a' times 'b' first, then multiply that result by 2.

MISTAKE: Applying the identity to (a-b)^2 instead of (a+b)^2. | CORRECTION: The identity for (a-b)^2 is slightly different (a^2 - 2ab + b^2). Make sure you use the correct identity for the given problem.

Practice Questions

Try It Yourself

QUESTION: Expand (x+4)^2. | ANSWER: x^2 + 8x + 16

QUESTION: If a = 7 and b = 2, find the value of (a+b)^2 using the identity. | ANSWER: 81

QUESTION: A square garden has sides of length (y+6) meters. What is the area of the garden? | ANSWER: y^2 + 12y + 36 square meters

MCQ

Quick Quiz

Which of these is the correct expansion of (p+q)^2?

p^2 + q^2

p^2 + 2pq + q^2

p^2 - 2pq + q^2

p + q^2

The Correct Answer Is:

The identity for (a+b)^2 is a^2 + 2ab + b^2. So, for (p+q)^2, it is p^2 + 2pq + q^2. Options A, C, and D are incorrect expansions of this identity.

Real World Connection

In the Real World

When city planners design a new residential complex, they often deal with square or rectangular plots. If a square plot has a side length of (road_width + building_length), they might use (a+b)^2 to quickly calculate the total area for planning and construction, ensuring enough space for homes and amenities like parks.

Key Vocabulary

Key Terms

IDENTITY: An equation that is true for all possible values of its variables. | EXPANSION: Writing an expression as a sum of its terms. | VARIABLE: A symbol, usually a letter, representing an unknown value. | SQUARE: Multiplying a number or expression by itself. | TERM: A single number, variable, or product of numbers and variables.

What's Next

What to Learn Next

Great job understanding (a+b) Whole Square! Next, you should explore "What is (a-b) Whole Square?". It's very similar but has a small, important difference, and knowing both will make you a pro at algebraic identities!