The given equation is:

$x^3 - 2x - 5 = 0$

This is a cubic equation, and solving it requires finding the value of $x$. There are several methods to approach this:

Step 1: Try Rational Root Theorem

The Rational Root Theorem suggests that if there is a rational root, it could be one of the factors of the constant term divided by the leading coefficient. In this case, the constant term is -5, and the leading coefficient is 1. So, the possible rational roots are:

$\pm 1, \pm 5$

Step 2: Test the possible roots

We'll substitute these values into the equation to check if any satisfy it:

1. $x = 1$: $(1)^3 - 2(1) - 5 = 1 - 2 - 5 = -6 \quad (\text{Not a root})$

2. $x = -1$: $(-1)^3 - 2(-1) - 5 = -1 + 2 - 5 = -4 \quad (\text{Not a root})$

3. $x = 5$: $(5)^3 - 2(5) - 5 = 125 - 10 - 5 = 110 \quad (\text{Not a root})$

4. $x = -5$: $(-5)^3 - 2(-5) - 5 = -125 + 10 - 5 = -120 \quad (\text{Not a root})$

Since none of the simple rational candidates work, the solution likely involves an irrational or complex root.

Step 3: Use Numerical Methods (Approximation)

One method to approximate the root is Newton's Method or using a calculator to find the real root. Based on known results, one real root of the equation $x^3 - 2x - 5 = 0$ is approximately:

$x \approx 2.094$

The other two roots are complex.

Would you like more details on how to apply numerical methods, or do you have any further questions?

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Related Questions:

1. How does Newton's Method work for solving cubic equations?
2. What are complex roots, and how can we find them for this equation?
3. What is the Rational Root Theorem, and why does it help in finding solutions?
4. Could we use factoring techniques to simplify similar cubic equations?
5. How do we graph $x^3 - 2x - 5$ to visualize the roots?

Tip: When solving cubic equations, sometimes transforming the equation or using approximation methods like Newton's Method provides faster solutions when exact factoring isn't possible.