Problem 64 - Intermediate Algebra

Question

Multiply and then simplify if possible. $$ (\sqrt{3 x}+2)(\sqrt{3 x}-2) $$

Step-by-Step Solution

Answer

The expression simplifies to \(3x - 4\).

Step 1: Identify the Formula

The expression you need to simplify is \[(\sqrt{3x}+2)(\sqrt{3x}-2)\].This resembles the difference of squares formula: \((a+b)(a-b) = a^2 - b^2\), where \(a = \sqrt{3x}\) and \(b = 2\).

Step 2: Apply the Difference of Squares Formula

Plug \(a = \sqrt{3x}\) and \(b = 2\) into the formula \(a^2 - b^2\):\[(\sqrt{3x})^2 - 2^2\].

Step 3: Simplify Each Term

Calculate \((\sqrt{3x})^2\) and \(2^2\):- \((\sqrt{3x})^2 = 3x\) because squaring a square root eliminates the root.- \(2^2 = 4\).

Step 4: Write Final Expression

Replace the squared terms back into the expression: \[3x - 4\]. This is the simplified form of the original expression.

Key Concepts

Multiplication of Radicals

Radicals can often seem daunting because of their roots, but multiplying them has a straightforward approach. When you have two radicals multiplied together, such as \[\sqrt{A} \times \sqrt{B}\], you can combine them under a single radical: \[\sqrt{A \times B}\]. However, this requires that both radicands, the numbers inside the radicals, are non-negative to simplify correctly without venturing into complex numbers.

When multiplying radicals of the same expression, for example, \[(\sqrt{3x} + 2)(\sqrt{3x} - 2)\], you use special formulas like the difference of squares rather than multiplying naively. This often makes the process quicker and less error-prone. Understanding the basics of multiplication of radicals helps to solve more complicated expressions efficiently.

Simplifying Expressions

Simplifying expressions involves reducing an expression to its simplest form. This typically means getting rid of any unnecessary elements such as parenthesis or terms that can be combined or otherwise simplified. When dealing with expressions such as\[3x - 4\],the expression is already simplified if there are no like terms or further operations that can be performed.

In our exercise, the simplification involves the concept of cancelling out terms or getting rid of radicals. For instance:

• Squaring a square root, \[(\sqrt{3x})^2\], gets rid of the root, leading to \[3x\].
• Subtracting constants, such as reducing \[4\], simplifies the expression further.

Each stage of simplification ensures that you are working with the most basic version of the expression.

Intermediate Algebra

Intermediate algebra blends foundational algebra with more complex topics, acting as a bridge to advanced studies. In this domain, you begin to see concepts like the difference of squares formula: \[(a+b)(a-b) = a^2 - b^2\]. This formula simplifies certain binomial products with ease, crucial in our exercise.

Here’s how this formula neatly simplifies expressions:

• Select the parts of the expression: \[(a+b)\] and \[(a-b)\]. In our case, it was \[a = \sqrt{3x}\] and \[b = 2\].
• Apply the formula: compute \[a^2\] and \[b^2\]. For instance: \[(\sqrt{3x})^2 = 3x\] and \[2^2=4\].
• Subtract these results for the final simplified expression: \[3x - 4\].

In intermediate algebra, understanding how to effectively deploy these formulas helps in tackling not just simple equations, but prepares you for calculus and beyond.