2%3Da2%2B2ab%2Bb2-examples.jpeg "(a+b)2=a2+2ab+b2 Examples")
Understanding the (a + b)2 Formula: A Detailed Explanation with Examples
The formula (a + b)2 = a2 + 2ab + b2 is a fundamental concept in algebra, used for expanding the square of a binomial. It states that squaring the sum of two terms is equal to the sum of the squares of each term plus twice the product of the two terms. Let's break it down and understand why it works.
Why (a + b)2 = a2 + 2ab + b2?
Imagine a square with sides of length (a + b). The area of this square would be (a + b)2. We can divide this square into smaller squares and rectangles:
- One square with sides of length 'a' (area = a2)
- One square with sides of length 'b' (area = b2)
- Two rectangles with sides of length 'a' and 'b' (area of each = ab)
The total area of the big square is the sum of the areas of these smaller figures: a2 + b2 + ab + ab. This simplifies to a2 + 2ab + b2. Therefore, (a + b)2 = a2 + 2ab + b2.
Examples of (a + b)2 = a2 + 2ab + b2:
Example 1:
- Problem: Expand (x + 3)2.
- Solution: Using the formula, we get:
- a = x
- b = 3
- (x + 3)2 = x2 + 2(x)(3) + 32
- (x + 3)2 = x2 + 6x + 9
Example 2:
- Problem: Simplify (2y + 5)2
- Solution:
- a = 2y
- b = 5
- (2y + 5)2 = (2y)2 + 2(2y)(5) + 52
- (2y + 5)2 = 4y2 + 20y + 25
Example 3:
- Problem: Find the value of (3m - 2)2
- Solution: This problem might look different, but we can still apply the formula.
- a = 3m
- b = -2
- (3m - 2)2 = (3m)2 + 2(3m)(-2) + (-2)2
- (3m - 2)2 = 9m2 - 12m + 4
Key Takeaways:
- The (a + b)2 formula is a powerful tool for expanding binomial expressions.
- Understanding the formula's derivation with the help of a visual representation helps solidify its application.
- Practice applying the formula with different expressions and variables for a deeper understanding of its use.