What is the Identity (a+b)^2?

S6-SA1-0041

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

The identity (a+b)^2 is a fundamental algebraic formula that tells us how to expand the square of a sum of two terms. It states that (a+b)^2 is equal to a^2 + 2ab + b^2. This identity is very useful for simplifying expressions and solving equations in mathematics.

Simple Example

Quick Example

Imagine you have two friends, Ajay (a) and Bhavesh (b), who each score some runs in a cricket match. If you want to find the square of their combined runs, you'd use (Ajay's runs + Bhavesh's runs)^2. The identity helps you quickly calculate this without just multiplying the total by itself, showing it's Ajay's runs squared, plus twice the product of their runs, plus Bhavesh's runs squared.

Worked Example

Step-by-Step

Let's expand (x+3)^2 using the identity (a+b)^2 = a^2 + 2ab + b^2.
---Step 1: Identify 'a' and 'b'. Here, a = x and b = 3.
---Step 2: Substitute 'a' and 'b' into the formula.
(x+3)^2 = (x)^2 + 2(x)(3) + (3)^2
---Step 3: Simplify each term.
(x)^2 = x^2
2(x)(3) = 6x
(3)^2 = 9
---Step 4: Combine the simplified terms.
(x+3)^2 = x^2 + 6x + 9
---Answer: x^2 + 6x + 9

Why It Matters

Understanding (a+b)^2 is crucial for fields like Engineering and Physics, where complex equations need simplifying. Engineers use it in designing structures and circuits, while Physicists apply it in calculating areas, volumes, and energy. It's a foundational tool for problem-solving in many scientific careers.

Common Mistakes

MISTAKE: Students often write (a+b)^2 = a^2 + b^2 | CORRECTION: Remember the middle term! The correct expansion is a^2 + 2ab + b^2.

MISTAKE: Incorrectly multiplying the middle term, for example, writing 2ab as 2a^2b or 2ab^2 | CORRECTION: The middle term is simply 2 times 'a' times 'b', so it's 2ab.

MISTAKE: Forgetting to square the numerical part of 'b' if 'b' is a number, e.g., (x+5)^2 = x^2 + 10x + 5 | CORRECTION: Square the number too. (x+5)^2 = x^2 + 10x + 25.

Practice Questions

Try It Yourself

QUESTION: Expand (y+7)^2 | ANSWER: y^2 + 14y + 49

QUESTION: Expand (2p+5)^2 | ANSWER: 4p^2 + 20p + 25

QUESTION: If (x+y)^2 = 100 and xy = 20, find x^2 + y^2. | ANSWER: 60

MCQ

Quick Quiz

Which of the following is the correct expansion of (3m+4)^2?

9m^2 + 16

3m^2 + 24m + 16

9m^2 + 12m + 16

9m^2 + 24m + 16

The Correct Answer Is:

Using (a+b)^2 = a^2 + 2ab + b^2, where a=3m and b=4. So, (3m)^2 + 2(3m)(4) + (4)^2 = 9m^2 + 24m + 16.

Real World Connection

In the Real World

This identity is used in computer graphics to calculate areas of shapes or distances between points on a screen, which helps in rendering realistic images for video games or animations. Even in designing simple structures like a square-shaped park with an extra border, this formula helps calculate the total area efficiently.

Key Vocabulary

Key Terms

IDENTITY: An equation that is true for all possible values of the variables | EXPAND: To multiply out the terms in an expression | TERM: A single number or variable, or numbers and variables multiplied together | SQUARE: To multiply a number or expression by itself

What's Next

What to Learn Next

Once you've mastered (a+b)^2, you're ready to explore (a-b)^2 and (a+b)(a-b). These identities are closely related and will further strengthen your algebraic problem-solving skills, opening doors to more complex mathematical challenges.