Problem 39 - Precalculus Mathematics for Calculus

Question

Multiply the algebraic expressions using a Special Product Formula and simplify. $$(\sqrt{x}+2)(\sqrt{x}-2)$$

Step-by-Step Solution

Answer

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

Step 1: Recognize the Special Product Formula

The expression \((\sqrt{x}+2)(\sqrt{x}-2)\) resembles the difference of squares formula, which is given by \((a+b)(a-b) = a^2 - b^2\). In the given problem, identify \(a\) as \(\sqrt{x}\) and \(b\) as 2.

Step 2: Apply the Difference of Squares Formula

Substitute \(a = \sqrt{x}\) and \(b = 2\) into the difference of squares formula. This gives us \((\sqrt{x})^2 - 2^2\).

Step 3: Simplify the Squares

Calculate the squares in the expression. \((\sqrt{x})^2\) simplifies to \(x\) and \(2^2\) simplifies to \(4\). Thus, the expression becomes \(x - 4\).

Key Concepts

Difference of Squares

The difference of squares is a handy algebraic technique that can simplify certain expressions quickly. This technique revolves around the special product formula \((a+b)(a-b) = a^2 - b^2\). This formula states that if you have two conjugated binomial expressions, meaning they have the same terms but opposite signs, you can multiply them using this formula.
In the problem \((\sqrt{x}+2)(\sqrt{x}-2)\), we see that it directly fits the structure of the difference of squares formula. Here, \(a\) is \(\sqrt{x}\) and \(b\) is 2.
Knowing the difference of squares formula not only helps you speed up calculations but also minimizes the chances of errors when multiplying expressions. Rather than expanding methodically, you apply the formula and get to the result in just one step. It represents a fundamental building block in algebra that simplifies the multiplication of certain binomials.

Algebraic Expressions

An algebraic expression constitutes numbers, variables, and operations (such as addition and multiplication). In this exercise, the expressions involved are \(\sqrt{x}+2\) and \(\sqrt{x}-2\). These are considered binomial expressions because they consist of two terms each.
When working with algebraic expressions, it's useful to understand how to manipulate and transform them using algebraic rules and formulas. This practice allows simplification and solving of algebraic equations.
The given problem involves radicals, specifically the square root of \(x\). Handling radicals can be tricky, as they require understanding how they interact with other algebraic operations. Familiarity with how radicals behave in multiplication, especially when conjugated, is a key skill in algebra.

Simplification

Simplifying algebraic expressions means reducing them to their simplest form. This involves executing operations in the right sequence and eliminating any unnecessary clutter, such as redundant terms.
In the final step of the exercise, we apply the simplification process to the expression \((\sqrt{x})^2 - 2^2\). Calculating these gives us \(x\) and 4, respectively. Substituting these simplifies the expression to \(x - 4\).
Simplification plays an essential role in making complex expressions more manageable and easier to interpret. It aids in bringing clarity and ensuring that the expressions are as clean and concise as possible. This ensures ease of understanding and further processing, especially useful in equations and problem-solving scenarios.