3. Consider the equation x² = √√x-1 a. Construct f(x) and sketch a graph of f(x) to illustrate the location of the positive root b. Find the root to 6 decimal places. Root: n Xn Xn+1
3. Consider the equation x² = √√x-1 a. Construct f(x) and sketch a graph of f(x) to illustrate the location of the positive root b. Find the root to 6 decimal places. Root: n Xn Xn+1
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter1: Expressions And Functions
Section1.8: Interpreting Graphs Of Functions
Problem 1GP
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Question
![**Educational Content: Solving Nonlinear Equations**
**Problem 3: Consider the equation \(x^2 = \sqrt{x - 1}\)**
**a. Construct \(f(x)\) and sketch a graph of \(f(x)\) to illustrate the location of the positive root.**
To solve this problem, we need to construct a function \(f(x)\) and then visualize it with a graph. The function can be formed as follows:
\[ f(x) = x^2 - \sqrt{x - 1} \]
We aim to find the value of \(x\) where \(f(x) = 0\), which represents the root of the equation. Graphing this function will help in estimating where the root lies.
**b. Find the root to 6 decimal places.**
To find the root accurately, we can use numerical methods such as the Newton-Raphson method, the bisection method, or another root-finding algorithm.
The table provided is used for iterative calculation:
\[
\begin{array}{|c|c|c|}
\hline
n & x_n & x_{n+1} \\
\hline
& & \\
\hline
& & \\
\hline
& & \\
\hline
& & \\
\hline
& & \\
\hline
& & \\
\hline
\end{array}
\]
**Root: ________________________**
This area is reserved for the final answer, which should be rounded to six decimal places after performing the necessary calculations.
---
**Note:** To solve this on an educational website, you would include a step-by-step guide on how to use the selected numerical method and show the progression within each iteration in the table above. This would teach students not only the result but also the process of iterative numerical solving.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F43e506db-8dc4-4921-8a99-ad8ac8b3c405%2F972a9924-7ae5-4958-a84a-80defd20f4bc%2F6rivu69_processed.png&w=3840&q=75)
Transcribed Image Text:**Educational Content: Solving Nonlinear Equations**
**Problem 3: Consider the equation \(x^2 = \sqrt{x - 1}\)**
**a. Construct \(f(x)\) and sketch a graph of \(f(x)\) to illustrate the location of the positive root.**
To solve this problem, we need to construct a function \(f(x)\) and then visualize it with a graph. The function can be formed as follows:
\[ f(x) = x^2 - \sqrt{x - 1} \]
We aim to find the value of \(x\) where \(f(x) = 0\), which represents the root of the equation. Graphing this function will help in estimating where the root lies.
**b. Find the root to 6 decimal places.**
To find the root accurately, we can use numerical methods such as the Newton-Raphson method, the bisection method, or another root-finding algorithm.
The table provided is used for iterative calculation:
\[
\begin{array}{|c|c|c|}
\hline
n & x_n & x_{n+1} \\
\hline
& & \\
\hline
& & \\
\hline
& & \\
\hline
& & \\
\hline
& & \\
\hline
& & \\
\hline
\end{array}
\]
**Root: ________________________**
This area is reserved for the final answer, which should be rounded to six decimal places after performing the necessary calculations.
---
**Note:** To solve this on an educational website, you would include a step-by-step guide on how to use the selected numerical method and show the progression within each iteration in the table above. This would teach students not only the result but also the process of iterative numerical solving.
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