To solve the equation \( x^3 + x - 1 = 0 \), we can use various methods including numerical methods, but there isn't a straightforward algebraic solution for this cubic equation. Here is a step-by-step approach:

### Analytical Approach (Cardano's Method)
1. **Standard Form**:
   The given equation is already in the standard form of a cubic equation \( ax^3 + bx^2 + cx + d = 0 \) where \( a = 1 \), \( b = 0 \), \( c = 1 \), and \( d = -1 \).

2. **Depressed Cubic**:
   To apply Cardano's method, we can convert the cubic equation into a depressed cubic form \( t^3 + pt + q = 0 \) using the substitution \( x = t - \frac{b}{3a} \). Here \( b = 0 \), so \( x = t \).

   The equation \( t^3 + t - 1 = 0 \) is already in the form \( t^3 + pt + q = 0 \) with \( p = 1 \) and \( q = -1 \).

3. **Solving the Depressed Cubic**:
   We use the formulas:
   \[
   t = \sqrt[3]{-\frac{q}{2} + \sqrt{\left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3}} + \sqrt[3]{-\frac{q}{2} - \sqrt{\left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3}}
   \]

   Substituting \( p = 1 \) and \( q = -1 \):
   \[
   t = \sqrt[3]{\frac{1}{2} + \sqrt{\left(\frac{1}{2}\right)^2 + \left(\frac{1}{3}\right)^3}} + \sqrt[3]{\frac{1}{2} - \sqrt{\left(\frac{1}{2}\right)^2 + \left(\frac{1}{3}\right)^3}}
   \]
   \[
   t = \sqrt[3]{\frac{1}{2} + \sqrt{\frac{1}{4} + \frac{1}{27}}} + \sqrt[3]{\frac{1}{2} - \sqrt{\frac{1}{4} + \frac{1}{27}}}
   \]
   \[
   t = \sqrt[3]{\frac{1}{2} + \sqrt{\frac{27 + 4}{108}}} + \sqrt[3]{\frac{1}{2} - \sqrt{\frac{27 + 4}{108}}}
   \]
   \[
   t = \sqrt[3]{\frac{1}{2} + \sqrt{\frac{31}{108}}} + \sqrt[3]{\frac{1}{2} - \sqrt{\frac{31}{108}}}
   \]

   Since the exact cubic roots are complex, the solution involves complex numbers and requires numerical approximation methods for practical purposes.

### Numerical Solution
Using numerical methods like Newton-Raphson or graphing, we can find the approximate roots.

### Graphical Method
We can plot the function \( f(x) = x^3 + x - 1 \) and find the roots by identifying where the graph intersects the x-axis.

### Approximate Solution
The real root can be approximated numerically:

Using a numerical solver (like Newton-Raphson or a calculator):
\[
x \approx 0.682327803828
\]

### Verification
We can verify this by substituting back into the original equation:
\[
(0.682327803828)^3 + 0.682327803828 - 1 \approx 0
\]

Thus, the solution to \( x^3 + x - 1 = 0 \) is approximately \( x \approx 0.6823 \).

Would you like more details or have any questions?

Here are 5 more questions you might be interested in:

1. How to solve quadratic equations?
2. What is the derivative of \( \sin(x) \)?
3. How to find the integral of \( e^x \)?
4. What are the roots of the equation \( x^2 - 4x + 4 = 0 \)?
5. How do you find the limit of a function as \( x \) approaches a specific value?

**Tip:** When solving polynomial equations, graphical methods can provide quick insights into the approximate locations of the roots.

Explore the solution of the cubic equation x^3 + x - 1 using analytical methods like Cardano's method and numerical methods. Includes step-by-step solutions, formula applications, and numerical approximations. Suitable for college-level mathematics.

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