%5E2-formula-class-9.jpeg "(x+y)^2 Formula Class 9")
Understanding the (x + y)² Formula in Class 9
The (x + y)² formula is a fundamental concept in algebra that plays a crucial role in simplifying expressions and solving equations. It's essential for students in Class 9 to grasp this formula thoroughly to excel in their math studies.
The Formula and Its Derivation
The (x + y)² formula states that:
(x + y)² = x² + 2xy + y²
This formula can be derived by expanding the expression (x + y)²:
(x + y)² = (x + y)(x + y)
Using the distributive property, we multiply each term in the first bracket by each term in the second bracket:
(x + y)(x + y) = x(x + y) + y(x + y)
Expanding further:
x(x + y) + y(x + y) = x² + xy + xy + y²
Combining like terms:
x² + xy + xy + y² = x² + 2xy + y²
Applying the Formula
The (x + y)² formula can be applied in various situations, including:
- Simplifying algebraic expressions: When encountering expressions like (a + b)², we can directly substitute the formula to get a² + 2ab + b² and simplify further.
- Solving equations: In equations containing squared terms, we can use the formula to rewrite them in a simpler form. For example, if we have the equation (x + 3)² = 16, we can apply the formula to get x² + 6x + 9 = 16 and solve for x.
- Factoring expressions: The formula can be used in reverse to factor expressions of the form x² + 2xy + y² as (x + y)².
Examples
Let's look at some examples to illustrate the application of the (x + y)² formula:
Example 1:
Simplify the expression (2a + 3b)².
Applying the formula:
(2a + 3b)² = (2a)² + 2(2a)(3b) + (3b)²
Expanding and simplifying:
= 4a² + 12ab + 9b²
Example 2:
Solve the equation (x + 5)² = 25.
Using the formula:
x² + 10x + 25 = 25
Subtracting 25 from both sides:
x² + 10x = 0
Factoring out x:
x(x + 10) = 0
Therefore, x = 0 or x = -10
Conclusion
The (x + y)² formula is a powerful tool in algebra that helps simplify expressions, solve equations, and factor expressions. Mastering this formula is crucial for success in Class 9 math and beyond. Through practice and application, students can gain a deep understanding of this fundamental concept and its various uses.