2%3Da2%2B2ab%2Bb2-examples.jpeg "(a+b)2=a2+2ab+b2 Examples")
(a+b)2: A Simple yet Powerful Algebraic Identity
In algebra, one of the most widely used identities is (a+b)2, which is equal to a2 + 2ab + b2. This identity is commonly used in various mathematical operations, such as expanding binomials, solving quadratic equations, and simplifying algebraic expressions.
What is (a+b)2?
The expression (a+b)2 is the square of the binomial a+b. It is calculated by multiplying the binomial by itself, resulting in:
(a+b)2 = (a+b)(a+b)
When we multiply the two binomials, we get:
(a+b)2 = a2 + ab + ba + b2
Simplifying the Expression
By combining like terms, we can simplify the expression to:
(a+b)2 = a2 + 2ab + b2
This is the standard form of the identity.
Examples
Let's look at some examples to illustrate how (a+b)2 works:
Example 1: Expanding (x+3)2
Using the identity, we get:
(x+3)2 = x2 + 2(x)(3) + 32 = x2 + 6x + 9
Example 2: Expanding (2y-4)2
Using the identity, we get:
(2y-4)2 = (2y)2 + 2(2y)(-4) + (-4)2 = 4y2 - 16y + 16
Example 3: Simplifying an Algebraic Expression
Simplify the expression:
x2 + 4x + 4
Using the identity, we can rewrite the expression as:
(x+2)2 = x2 + 2(x)(2) + 22 = x2 + 4x + 4
Conclusion
The (a+b)2 identity is a powerful tool in algebra, allowing us to expand binomials, simplify algebraic expressions, and solve quadratic equations with ease. By understanding and applying this identity, you'll become proficient in manipulating algebraic expressions and solving complex problems.
