Problem 73 - Elementary Algebra

Question

Find the product. \((2 x-7)(x+4)\).

Step-by-Step Solution

Answer

Answer: The product of the expressions (2x-7) and (x+4) is 2x^2 + x - 28.

Step 1: Distributive Property

First, multiply each term of the expression \((2x - 7)\) by each term of the expression \((x + 4)\). $$ (2x-7)(x+4) = (2x)(x) + (2x)(4) + (-7)(x) + (-7)(4) $$

Step 2: Simplify the terms

Now simplify each term. $$ (2x)(x) = 2x^2 $$ $$ (2x)(4) = 8x $$ $$ (-7)(x) = -7x $$ $$ (-7)(4) = -28 $$

Step 3: Combine and simplify the result

Combine the results from Step 2 and then simplify the expression. $$ 2x^2 + 8x - 7x - 28 = 2x^2 + (8x-7x) - 28 = 2x^2 + x - 28 $$ So the product of \((2x-7)(x+4)\) is: $$ (2x-7)(x+4) = 2x^2 + x - 28 $$

Key Concepts

Distributive Property

In the realm of algebra, the distributive property is a fundamental tool that helps in breaking down complex expressions. This property states that multiplication distributed over addition or subtraction. Simply put, it allows us to remove parentheses by distributing a multiplication across terms inside the parentheses. In our problem,

• (2x - 7) is multiplied by (x + 4).
• This means we take each term from the first parenthesis and multiply them individually by each term in the second parenthesis.

This results in:

• \((2x)(x)\),
• \((2x)(4)\),
• \((-7)(x)\),
• \((-7)(4)\).

By following this step, we transform the initial expression into a set of simpler terms, laying the foundation for further simplification.

Simplification of Expressions

Once we've applied the distributive property, our next step is to simplify the terms generated through distribution. Simplification involves carrying out basic arithmetic operations on these terms. This step becomes pivotal in reducing the equation into its most elementary form. Let's look at the simplified terms:

• \((2x)(x)\) results in \(2x^2\),
• \((2x)(4)\) simplifies to \(8x\),
• \((-7)(x)\) becomes \(-7x\),
• \((-7)(4)\) evaluates to \(-28\).

Here, each term is simplified using basic multiplication. By the end of this process, you have clear terms that can be further processed or combined. Simplification transforms the expression into an arrangement of standalone terms, making the algebraic equation much easier to understand.

Combining Like Terms

After simplification, the expression usually contains multiple terms that can be grouped together for further reduction. Combining like terms involves merging the terms with the same variable part. Think of it as adding or subtracting similar items in arithmetic. In our expression, we have

• \(8x\) and \(-7x\).

Both of these terms share the variable "x". To combine them, you add or subtract their coefficients:

• \(8x - 7x = x\).

Finally, our expression becomes:

• \(2x^2 + x - 28\).

By combining like terms, you not only simplify the equation further, but also make it more comprehensible and easier to handle in future calculations. This final expression tells you that your product has been fully simplified, with no comparable terms left to combine.