Problem 95 - College Algebra with Modeling and Visualization

Question

Multiply and simplify. $$ (\sqrt{x}-7)(\sqrt{x}+8) $$

Step-by-Step Solution

Answer

The simplified expression is \(x - 49\).

Step 1: Identify the Pattern

The expression \((\sqrt{x}-7)(\sqrt{x}+8)\) is in the form of a difference of squares \((a-b)(a+b) = a^2 - b^2\). Here \(a = \sqrt{x}\) and \(b = 7\).

Step 2: Apply the Difference of Squares Formula

Using the formula \((a-b)(a+b) = a^2 - b^2\), substitute \(a = \sqrt{x}\) and \(b = 7\). The expression becomes \((\sqrt{x})^2 - 7^2\).

Step 3: Simplify the Squares

Calculate each square: \((\sqrt{x})^2 = x\) and \(7^2 = 49\). Substitute these values back into the expression to get \(x - 49\).

Key Concepts

Difference of Squares

The difference of squares is a powerful algebraic tool. It's used when you have two binomials with a specific structure:

• One binomial involves subtracting two terms.
• The other involves adding the same two terms.

When you multiply these two binomials together, you get a simple result. In our case, the expression i \( (\sqrt{x}-7)(\sqrt{x}+8) \) fits this pattern where the terms in common are \(\sqrt{x} \) and 7. The multiplication follows this formula: \((a-b)(a+b) = a^2 - b^2\).Understanding this pattern allows you to quickly simplify the expression into a simpler form by squaring the unique terms.

Simplification

Simplification in algebra reduces expressions to their simplest form. This often makes calculations easier. In the difference of squares discussed earlier, the expression \((\sqrt{x})^2 - 7^2\) translates directly to \(x - 49\). Here's how you properly simplify such expressions:

• Identify the squares of each component. Here, \((\sqrt{x})^2\) becomes \(x\) and \(7^2\) becomes \(49\).
• Substitute these squared terms back into the main expression.
• Your final simplified expression is \(x - 49\).

Simplification removes complexity, letting you work with expressions more easily and accurately.

Radicals

Radicals, such as the square root, are expressions that involve roots. The power of a radical can transform expressions significantly. In the original exercise, we encountered a square root, \(\sqrt{x}\), which served as a key element in the difference of squares. To simplify expressions that contain radicals:

• Remember that the square of a square root returns the original value: \((\sqrt{x})^2 = x\).
• Use this property whenever simplifying expressions with radicals.

This knowledge of radicals aids in algebraic simplifications, particularly when paired with other concepts like the difference of squares. It's all about recognizing patterns and knowing how to handle each component step by step.