Expanding Using Identities

Algebra is a fascinating branch of mathematics that deals with symbols and the rules for manipulating those symbols. In this lesson, we will explore a fundamental aspect of algebra known as "expanding using identities." Students often encounter this concept when learning algebraic expressions. Expanding means rewriting an expression in expanded form. Algebraic identities simplify the process by providing ready-made formulas that help us expand expressions quickly and accurately. By understanding and using these identities, we can solve algebraic problems more efficiently.

Understanding algebraic identities

Algebraic identities are equations that are true for any value of the variables involved. They are like special tools that help us transform and simplify expressions. These identities are used to expand expressions, factor expressions, and even solve equations. Let's take a look at some common algebraic identities that are usually introduced in Class 7 Maths.

Some basic algebraic identities

• (a + b)^2 = a^2 + 2ab + b^2
• (a - b)^2 = a^2 - 2ab + b^2
• a^2 - b^2 = (a + b)(a - b)
• (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
• (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3

Process of expansion using identities

Expanding an expression means writing it in expanded form by removing the parentheses. Algebraic identities help us do this quickly. Let's see how we can use these identities to simplify and understand algebraic expressions through explanations, examples, and visualizations.

Example 1: (x + 3)^2 expansion

We use the identity (a + b)^2 = a^2 + 2ab + b^2.

    In this case, a = x and b = 3.
    So, (x + 3)^2 = x^2 + 2*x*3 + 3^2
    Which simplifies to x^2 + 6x + 9.

Example 2: (2y - 5)^2 expansion

We use the identity (a - b)^2 = a^2 - 2ab + b^2.

    Here, a = 2y and b = 5.
    So, (2y - 5)^2 = (2y)^2 - 2 * 2y * 5 + 5^2
    Which gives us 4y^2 - 20y + 25.

Imagination in detail

Algebraic extensions can also be visualized using geometric representations, where the areas of squares and rectangles correspond to the terms of the identities.

Visual example 1: (a + b)^2 expansion

The combined area is a² + 2ab + b².

Visual example 2: (a - b)^2 expansion

The combined area is a² - 2ab + b².

Further applications and examples

Expanding using identities not only helps to simplify expressions, but it is also essential when solving algebraic equations and problems in geometry, physics, and other subjects.

Example 3: (p + q)^3 expansion

Use the identity (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3.

    Here, a = p and b = q.
    So, (p + q)^3 = p^3 + 3p^2q + 3pq^2 + q^3.

Example 4: Using a^2 - b^2 = (a + b)(a - b)

    Given the expression x^2 - 16, note that it fits the form a^2 - b^2.
    Here, a = x and b = 4.
    Thus, x^2 - 16 = (x + 4)(x - 4).

Practice problems

Here are some practice problems to solidify your understanding of expanding using identities:

1. Expand (m + 7)^2.
2. Use identities to expand (3x - 2y)^2.
3. Find the expanded form of (a - 5b)^3.
4. Use identities to verify that 25x^2 - 9y^2 = (5x + 3y)(5x - 3y).

Conclusion

Algebraic identities provide us with quick and reliable ways to expand algebraic expressions. Understanding these identities not only helps solve mathematical problems more efficiently but also forms a strong foundation for advanced mathematical concepts. By practicing and visualizing these expansions, students can develop a deeper understanding of algebra and its applications.

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