What is (a+b) Squared?

S3-SA1-0749

Grade Level:

Class 6

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

Definition

What is it?

When we say "(a+b) squared," it means we are multiplying the entire expression (a+b) by itself. It's written as (a+b)^2. This concept helps us find the area of a square whose side length is represented by 'a + b'.

Simple Example

Quick Example

Imagine you have a square plot of land for your garden. If one side of the plot is 5 meters + 2 meters long, then the total length is (5+2) = 7 meters. To find the area of this square garden, you would calculate (5+2)^2, which is 7^2 = 49 square meters.

Worked Example

Step-by-Step

Let's expand (3+4) squared.

STEP 1: Understand (3+4)^2 means (3+4) multiplied by (3+4).

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STEP 2: First, simplify inside the bracket: (3+4) = 7.

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STEP 3: Now, square the result: 7^2.

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STEP 4: Calculate 7 multiplied by 7.

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STEP 5: 7 * 7 = 49.

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ANSWER: So, (3+4)^2 = 49.

Why It Matters

Understanding (a+b) squared is a basic building block for advanced math used in many fields. Engineers use it to design buildings and bridges, while computer scientists use similar ideas in coding and algorithms. It's also foundational for understanding concepts in physics and even economics, helping you analyze growth and changes.

Common Mistakes

MISTAKE: Students often write (a+b)^2 as a^2 + b^2 | CORRECTION: Remember to multiply (a+b) by (a+b) completely. It's not just squaring 'a' and squaring 'b' separately. The correct expansion is a^2 + 2ab + b^2.

MISTAKE: Forgetting the middle term '2ab' when expanding (a+b)^2 | CORRECTION: Always remember the formula: (a+b)^2 = a^2 + 2ab + b^2. The '2ab' term comes from multiplying 'a' by 'b' twice (once from each bracket).

MISTAKE: Confusing (a+b)^2 with (a-b)^2 or a^2 + b^2 | CORRECTION: Each algebraic identity is distinct. (a+b)^2 has a positive '2ab' term, while (a-b)^2 has a negative '2ab' term. a^2 + b^2 cannot be simplified further as a product.

Practice Questions

Try It Yourself

QUESTION: What is (6+1) squared? | ANSWER: 49

QUESTION: If a = 5 and b = 3, what is (a+b) squared? | ANSWER: 64

QUESTION: A square field has a side length of (7 + 3) meters. What is its area in square meters? | ANSWER: 100 square meters

MCQ

Quick Quiz

Which of the following expressions is equal to (x+y)^2?

x^2 + y^2

x^2 + 2xy + y^2

x^2 - y^2

2x + 2y

The Correct Answer Is:

The correct expansion of (x+y)^2 is x^2 + 2xy + y^2. Options A and C are common mistakes, while D is just adding the terms, not squaring.

Real World Connection

In the Real World

Imagine you are planning to tile a square room. If the room's side length is made up of two parts, say 4 meters plus 1 meter, you can use (4+1)^2 to quickly calculate the total area to buy tiles for. This is like how architects and construction workers calculate material needs for different sections of a building project.

Key Vocabulary

Key Terms

SQUARED: A number or expression multiplied by itself | EXPRESSION: A combination of numbers and variables with mathematical operations | ALGEBRAIC IDENTITY: An equation that is true for all values of the variables | TERM: A single number or variable, or numbers and variables multiplied together

What's Next

What to Learn Next

Great job understanding (a+b) squared! Next, you should explore "What is (a-b) squared?" and "What is (a^2 - b^2)?". These are also important algebraic identities that build on what you've learned here and will help you solve even more complex problems.