Problem 58 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to approximate any solutions of the equation. [Remember to write the equation in the form \(f(x)=0.1\) $$\sqrt{x-4}=8$$

Step-by-Step Solution

Answer

The solution to the equation \(\sqrt{x-4}=8\) is \(x=68\).

Step 1: Write the equation in the form \(f(x)=0\)

An equation in the form \(f(x)=0\) can be suitably graphed. Our current equation is \(\sqrt{x-4}=8\). Squaring both sides removes the square root, resulting to \(x-4=64\). We finally transpose it to become \(x-64-4=0\), i.e., \(x-68=0\).

Step 2: Use a graphing utility to find the root

Input the function \(x-68=0\) into a graphing utility such as Wolfram Alpha, GeoGebra, or a graphing calculator. The root of the equation is the value of \(x\) at which the graph intersects the x-axis. In this case, the graph intersects the x-axis at 68. Therefore, the solution to the equation is \(x=68\).

Key Concepts

Roots of Equations

The concept of finding the roots of equations is fundamental in solving mathematical problems. A root of an equation is simply the value of the variable that makes the equation true. In other words, when you substitute this value into the equation, you should get zero. For example, in the equation \(x-68=0\), the root is \(x=68\). This is because when you substitute 68 into the equation, you get \(68 - 68 = 0\).

Finding the roots of equations often involves manipulation, either through algebraic methods or using graphical solutions. Identifying roots helps in understanding the behavior and attributes of functions, whether linear or complex.

Function Transformation

Function transformation involves changing the form of a function to make it easier to work with. This could include operations such as squaring, taking the square root, or transposing terms. In the given exercise, we started with the equation \(\sqrt{x-4}=8\). To simplify, we squared both sides to remove the square root, transforming it into \(x-4=64\).

This simplification makes it much easier to solve complex expressions. Understanding function transformations is key to manipulating functions' forms and is a useful skill in both algebra and calculus.

• Squaring both sides of an equation can help remove radicals.
• Manipulating terms help to isolate variables.

Use of Graphing Calculators

Using graphing calculators is an effective way to find roots and analyze functions. Once an equation is in the form \(f(x) = 0\), a graphing calculator can quickly show where the graph intersects the x-axis. These intersections are the roots of the equation.

For example, with the function \(x-68=0\), inputting this into a graphing calculator will visually display the root as the point where the graph crosses the x-axis at \(x=68\).

• Graphing calculators can handle complex equations easily and provide visual aids.
• They allow for checking work on complex algebraic manipulations quickly.

Graphing calculators streamline the entire process of identifying intersections, making solving equations more straightforward.

Mathematical Transposition

Mathematical transposition is a technique used to rearrange equations to isolate variables or simplify expressions. This method becomes particularly useful when solving equations manually. In the original problem, after removing the square root by squaring both sides, we rearranged \(x-4=64\) by adding 4 to both sides to isolate \(x\).

This resulted in the equation \(x = 68\). By transposing, you convert an equation into a more solvable form, essentially shifting elements across the equation sign while changing their operations.

• Transposition helps in isolating variables to one side of the equation.
• It's crucial for simplifying equations before using any graphing utilities.

Understanding how to transpose is a fundamental skill that simplifies and optimizes solving equations.