Simplifying Expressions Multiply -3a(a+2) Using The Distributive Property
Understanding the Distributive Property
In mathematics, the distributive property is a fundamental concept that allows us to simplify expressions involving multiplication and addition or subtraction. This property is especially useful when dealing with algebraic expressions that contain parentheses. In essence, the distributive property states that for any real numbers a, b, and c, the following equation holds true:
a(b + c) = ab + ac
This means that we can multiply the term outside the parentheses (a) by each term inside the parentheses (b and c) separately, and then add the results. Similarly, for subtraction, the distributive property can be expressed as:
a(b - c) = ab - ac
Here, we multiply 'a' by both 'b' and 'c', but since 'c' is being subtracted, we subtract the product of 'a' and 'c' from the product of 'a' and 'b'. Understanding and applying the distributive property is crucial for simplifying algebraic expressions, solving equations, and various other mathematical operations. It allows us to break down complex expressions into simpler, more manageable parts, making it easier to perform calculations and arrive at the correct solution. Without the distributive property, many algebraic manipulations would be significantly more challenging, and it serves as a cornerstone for more advanced mathematical concepts.
When tackling mathematical problems, the distributive property often arises in situations where we need to expand expressions or eliminate parentheses. For instance, consider the expression 3(x + 2). To simplify this, we distribute the 3 to both the x and the 2, resulting in 3 * x + 3 * 2, which simplifies to 3x + 6. This simple example illustrates the power of the distributive property in transforming expressions into a more workable form. In more complex scenarios, the distributive property might be combined with other algebraic techniques, such as combining like terms or factoring, to fully simplify an expression or solve an equation. Mastering the distributive property is therefore an essential step in building a strong foundation in algebra and mathematics in general.
Furthermore, the distributive property is not limited to simple expressions involving numbers and variables. It can also be applied to more complex expressions involving polynomials, fractions, and even functions. For example, if we have the expression (x + 1)(x + 2), we can use the distributive property (sometimes referred to as the FOIL method in this specific case) to expand it: (x + 1)(x + 2) = x(x + 2) + 1(x + 2) = x^2 + 2x + x + 2 = x^2 + 3x + 2. This demonstrates how the distributive property can be extended to handle expressions with multiple terms. Similarly, when dealing with fractions, the distributive property can be used to simplify expressions such as (1/2)(4x + 6), which simplifies to 2x + 3. The versatility of the distributive property makes it an indispensable tool in various mathematical contexts, and a thorough understanding of its applications is crucial for success in algebra and beyond.
Applying the Distributive Property to the Given Expression
To simplify the expression
-3a(a + 2)
we will use the distributive property. The distributive property, as discussed earlier, allows us to multiply a term outside parentheses by each term inside the parentheses. In this case, we have -3a outside the parentheses and (a + 2) inside the parentheses.
Following the distributive property, we multiply -3a by each term inside the parentheses:
-3a * a + (-3a) * 2
Now, let's perform each multiplication separately.
Multiplying -3a by a
When multiplying variables with exponents, we add the exponents. In this case, -3a is the same as -3a^1, and a is the same as a^1. So, when we multiply -3a^1 by a^1, we get:
-3a^1 * a^1 = -3a^(1+1) = -3a^2
Multiplying -3a by 2
Here, we are multiplying a constant (-3) by another constant (2) and a variable (a). The multiplication of the constants is straightforward:
-3 * 2 = -6
So, the result of multiplying -3a by 2 is:
-6a
Combining the Results
Now that we have performed both multiplications, we can combine the results to get the simplified expression. We found that:
- -3a * a = -3a^2
- -3a * 2 = -6a
Adding these two results together, we get:
-3a^2 + (-6a)
Which simplifies to:
-3a^2 - 6a
Therefore, the simplified form of the expression -3a(a + 2) is -3a^2 - 6a. This is the final answer, as there are no more like terms to combine and the expression is in its simplest form. Understanding and applying the distributive property correctly is essential for simplifying algebraic expressions and solving equations, and this example illustrates a clear step-by-step process for doing so.
This process demonstrates how the distributive property allows us to break down complex expressions into simpler terms, making it easier to manipulate and solve them. By multiplying each term inside the parentheses by the term outside, we eliminate the parentheses and can then combine like terms if necessary. In this case, the resulting expression, -3a^2 - 6a, is a quadratic expression, which has different properties and behaviors compared to the original expression with parentheses. The ability to move between these forms is a critical skill in algebra and higher-level mathematics.
Moreover, simplifying expressions is not just about finding a shorter or more compact form. It is also about revealing the underlying structure of the expression and making it easier to analyze and use in further calculations. For instance, the simplified expression -3a^2 - 6a might be used in a quadratic equation or as part of a larger algebraic problem. Being able to quickly and accurately simplify expressions using the distributive property is a fundamental skill that will support more advanced mathematical work.
Final Answer
Therefore, after applying the distributive property and simplifying, the final answer is:
-3a^2 - 6a
