%5E2-expanded.jpeg "(a+b)^2 Expanded")
Understanding (a + b)^2: The Square of a Binomial
The expression (a + b)^2 is a common one in algebra, representing the square of a binomial. A binomial is a polynomial with two terms, in this case, "a" and "b". Understanding how to expand this expression is crucial for various algebraic manipulations and problem-solving.
The Expansion:
(a + b)^2 can be expanded as follows:
(a + b)^2 = (a + b)(a + b)
To expand this, we apply the distributive property:
- First term: a * a = a^2
- Outer term: a * b = ab
- Inner term: b * a = ba (which is the same as ab)
- Last term: b * b = b^2
Adding these terms together, we get:
(a + b)^2 = a^2 + ab + ba + b^2
Finally, combining the like terms:
(a + b)^2 = a^2 + 2ab + b^2
Key Takeaways:
- The square of a binomial results in a trinomial (an expression with three terms).
- The middle term is always twice the product of the two terms in the binomial.
- The expansion can be remembered as the "FOIL" method: First, Outer, Inner, Last.
Examples:
Let's look at some examples to illustrate the expansion:
- (x + 2)^2
Using the formula:
(x + 2)^2 = x^2 + 2(x)(2) + 2^2 = x^2 + 4x + 4
- (3y - 1)^2
(3y - 1)^2 = (3y)^2 + 2(3y)(-1) + (-1)^2 = 9y^2 - 6y + 1
Applications:
Expanding (a + b)^2 is a fundamental concept used in:
- Factoring quadratics: Recognizing the pattern helps in factoring expressions like x^2 + 6x + 9.
- Solving equations: Expanding the square allows you to simplify and solve equations involving binomials.
- Simplifying expressions: It's crucial for simplifying algebraic expressions, especially when dealing with complex polynomials.
Understanding the expansion of (a + b)^2 is a key step in building a solid foundation in algebra. By applying the formula and practicing examples, you can master this important algebraic concept and confidently tackle more complex problems.