Problem 69 - Algebra 1

Question

COMBINING LIKE TERMS Apply the distributive property. Then simplify by combining like terms. $$ 4 w^{2}-w(2 w-3) $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(2w^{2} - 3w\).

Step 1: Apply the Distributive Property

To apply the distributive property to the term \(w(2w - 3)\), multiply \(w\) by each of the terms inside the parentheses. This yields \(w*2w - w*3\), or \(2w^{2} - 3w\).

Step 2: Substitute into the Original Expression

Now substitute \(2w^{2} - 3w\) for \(w(2w - 3)\) in the original expression to obtain \(4w^{2} - (2w^{2} - 3w)\).

Step 3: Simplification and Combining Like Terms

Simplify this expression and combine like terms. There are two terms that involve \(w^{2}\) and one term that involves \(w\). First, combine the terms that involve \(w^{2}\): \(4w^{2} - 2w^{2} = 2w^{2}\). Note that the \(3w\)-term remains unchanged, yielding the final expression: \(2w^{2} - 3w\).

Key Concepts

Distributive Property

When working with algebra, the distributive property is a fundamental rule that allows us to multiply a single term by each term within a parenthesis. For instance, in the expression \(a(b + c)\), the distributive property enables us to 'distribute' the multiplication of \(a\) to both \(b\) and \(c\), resulting in \(ab + ac\). This property simplifies complicated algebraic expressions and makes it easier to solve equations.

Now, applying this to our problem: to simplify the expression \(w(2w - 3)\), we distribute the \(w\) across the terms inside the parenthesis, multiplying it by both \(2w\) and \( -3\). This results in the expression \(2w^2 - 3w\), effectively breaking down the larger problem into more manageable parts.

Simplify Expressions

To simplify an expression means to make it as straightforward and condensed as possible. Simplification is like clearing the clutter from a room to see everything more clearly. In algebra, this might involve combining like terms, reducing fractions, or applying mathematical properties like the distributive property mentioned earlier.

Combining like terms is essential in simplification. Like terms have the same variables raised to the same powers. For example, \(2w^2\) and \(4w^2\) are like terms because they both contain the variable \(w\) squared. We can combine them by adding or subtracting their coefficients (the numbers in front of the variables), which in our exercise results in \(4w^2 - 2w^2 = 2w^2\). Keeping our expressions neat and orderly not only helps with clarity but also sets us up for success in finding the right solution.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables (like \(x\) or \(y\)), and operation signs (like \(+\), \(-\), \(*\), and \(\div\)). They represent quantities that can vary, which is why we use variables. Think of algebraic expressions as sentences where the quantities tell a story - their relationships and interactions unfold through the operations that connect them.

Understanding how to manipulate these expressions is crucial for solving algebra problems. Our initial expression, \(4w^2-w(2w-3)\), is an example that, through manipulation using the distributive property and combining like terms, ultimately simplifies to \(2w^2 - 3w\). This tidier expression still holds all the information of the initial one but in a more streamlined form that is easier to evaluate or further manipulate in solving equations.