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.