Find all real x satisfying x=sqrt(x-(1/x))+sqrt(1-(1/x)).
So far I tried to isolate the roots and solve it analytically, to which I got the answer 1, which doesn't work when plugged in

To solve the equation \( x = \sqrt{x - \frac{1}{x}} + \sqrt{1 - \frac{1}{x}} \), we need to be careful with our approach since isolating the roots might not work easily.

Let’s proceed step by step to analyze the equation:

### Step 1: Investigate the domain
We know that both square roots must be defined, meaning the expressions inside must be non-negative.

1. \( \sqrt{x - \frac{1}{x}} \) is defined when \( x - \frac{1}{x} \geq 0 \). This implies:
   \[
   x \geq \frac{1}{x} \quad \text{or equivalently} \quad x^2 \geq 1.
   \]
   Hence, \( x \geq 1 \) or \( x \leq -1 \).

2. \( \sqrt{1 - \frac{1}{x}} \) is defined when \( 1 - \frac{1}{x} \geq 0 \). This implies:
   \[
   x \geq 1.
   \]
   Thus, \( x \geq 1 \) is the only domain where both square roots are defined.

### Step 2: Test possible solutions
The domain suggests that \( x \geq 1 \). Let’s test specific values of \( x \) starting from simple candidates:

#### Test \( x = 1 \):
Substitute \( x = 1 \) into the original equation:
\[
1 = \sqrt{1 - \frac{1}{1}} + \sqrt{1 - \frac{1}{1}} = \sqrt{1 - 1} + \sqrt{1 - 1} = 0 + 0 = 0.
\]
This is false, so \( x = 1 \) is not a solution.

#### Test \( x = 2 \):
Substitute \( x = 2 \) into the original equation:
\[
2 = \sqrt{2 - \frac{1}{2}} + \sqrt{1 - \frac{1}{2}}.
\]
Let’s calculate each term:
\[
\sqrt{2 - \frac{1}{2}} = \sqrt{2 - 0.5} = \sqrt{1.5} \approx 1.2247,
\]
\[
\sqrt{1 - \frac{1}{2}} = \sqrt{1 - 0.5} = \sqrt{0.5} \approx 0.7071.
\]
So, the right-hand side is approximately:
\[
1.2247 + 0.7071 = 1.9318,
\]
which is close to 2. Thus, \( x = 2 \) is approximately a solution.

#### Test \( x = 3 \):
Substitute \( x = 3 \) into the original equation:
\[
3 = \sqrt{3 - \frac{1}{3}} + \sqrt{1 - \frac{1}{3}}.
\]
Let’s calculate each term:
\[
\sqrt{3 - \frac{1}{3}} = \sqrt{3 - 0.3333} = \sqrt{2.6667} \approx 1.632,
\]
\[
\sqrt{1 - \frac{1}{3}} = \sqrt{1 - 0.3333} = \sqrt{0.6667} \approx 0.8165.
\]
So, the right-hand side is approximately:
\[
1.632 + 0.8165 = 2.4485,
\]
which is far from 3, so \( x = 3 \) is not a solution.

### Step 3: Conclude the solution
From these tests, we can conclude that \( x = 2 \) is the only real solution that satisfies the given equation approximately.

Would you like a detailed explanation of any part of the steps, or further clarification?

### Related Questions:
1. Can you derive a general method for solving equations involving square roots like this one?
2. What methods can be used to estimate the error in approximate solutions?
3. How would this problem change if negative values of \( x \) were considered?
4. Can you confirm the solution using graphical methods?
5. Is there a way to approach this problem using inequalities instead?

### Tip:
When solving equations with square roots, always carefully consider the domain of the function before trying to solve the equation directly.

Find all real x satisfying x=sqrt(x-(1/x))+sqrt(1-(1/x)). So far I tried to isolate the roots and solve it analytically, to which I got the answer 1, which doesn't work when plugged in.

This page provides a detailed solution to the radical equation x = sqrt(x - (1/x)) + sqrt(1 - (1/x)). The solution involves analyzing the domain of square root expressions and testing values to find an approximate real solution. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation