Solving the Quadratic Equation: (x^2 2x 1 0) — Methods and Applications

In this article, we will explore how to solve the quadratic equation (x^2 2x 1 0). We will discuss different methods to solve this equation, including factoring, using the quadratic formula, and understanding the role of complex numbers. We will also explore the significance of the solutions.

Factoring Method

The given equation can be factored as follows:

Rewrite the equation: (x^2 2x 1 0) Recognize the perfect square: (x^2 2x 1 (x 1)^2) Set the factor to zero: (x 1 0) Solve for (x): (x -1)

Note: The equation (x^2 2x 1 0) can be rewritten as (x^2 - (-2x - 1) 0), which clearly shows the factorization.

Quadratic Formula Method

Alternatively, you can use the quadratic formula to solve the equation. The quadratic formula is given by:

(x frac{-b pm sqrt{b^2 - 4ac}}{2a})

For the equation (x^2 2x 1 0), we have:

(a 1) (b 2) (c 1)

Calculate the discriminant:

(b^2 - 4ac 2^2 - 4 cdot 1 cdot 1 4 - 4 0)

Apply the quadratic formula:

(x frac{-2 pm sqrt{0}}{2 cdot 1} frac{-2 pm 0}{2} frac{-2}{2} -1)

Therefore, the solution to the equation (x^2 2x 1 0) is (x -1).

Complex Number Solutions

When the discriminant is negative, we introduce complex numbers to find solutions. In the equation (x^2 - 1 0), we have:

(x^2 -1)

We define (i) such that (i^2 -1). Therefore, the solutions are:

(x pm i)

This is generally introduced in the first year of university mathematics. Such solutions have applications in various fields, including physics, engineering, and advanced mathematics.

Graphical Method

You can also verify the solution by graphing the equation (y x^2 2x 1). The x-intercept of the parabola, where the graph touches the x-axis, is the solution. For this equation, the graph is a perfect square with the x-intercept at (x -1).

Conclusion

In this article, we have explored various methods to solve the quadratic equation (x^2 2x 1 0). The solution is (x -1). Whether using factoring, the quadratic formula, or introducing complex numbers, we have found a single solution, which is a real number. This solution is significant and can be extended to more complex problems involving complex numbers.

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