x1^{3}+2x1^{2}+x1-3=x2^{3}+2x2^{2}+x2-3 iguala x1=x2

To solve the equation \(x_1^3 + 2x_1^2 + x_1 - 3 = x_2^3 + 2x_2^2 + x_2 - 3\) and find when \(x_1 = x_2\), we can directly substitute \(x_1 = x_2\) into the equation. This simplifies the equation and removes \(x_2\).

### Step-by-Step Solution:

1. Start with the given equation:

   \[
   x_1^3 + 2x_1^2 + x_1 - 3 = x_2^3 + 2x_2^2 + x_2 - 3
   \]

2. Substitute \(x_1 = x_2\) into the equation:

   \[
   x_1^3 + 2x_1^2 + x_1 - 3 = x_1^3 + 2x_1^2 + x_1 - 3
   \]

3. Simplify both sides:

   \[
   x_1^3 + 2x_1^2 + x_1 - 3 = x_1^3 + 2x_1^2 + x_1 - 3
   \]

4. Since both sides of the equation are exactly the same, the equation holds true for all values of \(x_1\) and \(x_2\) as long as \(x_1 = x_2\).

### Conclusion:
The equation is true for any value of \(x_1\) when \(x_1 = x_2\). There are no specific restrictions on \(x_1\) and \(x_2\), meaning any real number solution for \(x_1 = x_2\) will satisfy the equation.

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Would you like further details or have any questions?

Here are 5 questions that expand on the concept:

1. What happens if we introduce new terms to the equation, like \(x_1^4\)?
2. How would the solution change if there was a different constant instead of \(-3\)?
3. How can we visualize this equality graphically?
4. What methods can be used to solve if the equation were not symmetrical?
5. How does this concept apply to systems of equations?

**Tip:** When simplifying equations, always check for symmetry or identical terms on both sides. This can help you identify if they hold for all values.

x1^3 + 2x1^2 + x1 - 3 = x2^3 + 2x2^2 + x2 - 3 and solve for when x1 = x2

Explore the step-by-step solution for the cubic equation x1^3 + 2x1^2 + x1 - 3 = x2^3 + 2x2^2 + x2 - 3, where x1 equals x2. This problem demonstrates how to simplify cubic equations and check for symmetry, suitable for high school students in Grades 9-12.

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