Roots Of X 2 X 1

Solving the Quadratic Equation: x² + 2x + 1 = 0

This article explores the solution to the quadratic equation x² + 2x + 1 = 0, explaining the different methods available and providing a comprehensive understanding of the underlying mathematical principles. We will delve into factoring, the quadratic formula, and completing the square, offering a versatile approach to solving this specific equation and similar quadratic problems. This will help you understand not just the answer, but the why behind it.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (x in this case) is 2. They generally take the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. Our specific equation, x² + 2x + 1 = 0, has a = 1, b = 2, and c = 1. Understanding these coefficients is crucial for choosing the most efficient solution method.

Method 1: Factoring

Factoring is often the quickest method for solving quadratic equations, especially when the equation can be easily factored. In this case, x² + 2x + 1 is a perfect square trinomial. This means it can be factored into the square of a binomial:

(x + 1)(x + 1) = 0 or (x + 1)² = 0

This simplifies the solution considerably. To find the roots, we set each factor equal to zero:

x + 1 = 0

Solving for x, we get:

x = -1

Therefore, the equation x² + 2x + 1 = 0 has only one real root, x = -1. This is because the quadratic represents a parabola that touches the x-axis at only one point.

Method 2: The Quadratic Formula

The quadratic formula is a more general method that works for all quadratic equations, even those that are difficult or impossible to factor. The formula is:

x = [-b ± √(b² - 4ac)] / 2a

Plugging in the values from our equation (a = 1, b = 2, c = 1), we get:

x = [-2 ± √(2² - 4 * 1 * 1)] / (2 * 1) x = [-2 ± √(4 - 4)] / 2 x = [-2 ± √0] / 2 x = -2 / 2 x = -1

Again, we find that the only root is x = -1.

Method 3: Completing the Square

Completing the square is another powerful technique for solving quadratic equations. It involves manipulating the equation to create a perfect square trinomial, which can then be easily factored.

Starting with x² + 2x + 1 = 0:

1. Move the constant term to the right side: x² + 2x = -1

2. Take half of the coefficient of x (which is 2), square it (2/2 = 1, 1² = 1), and add it to both sides: x² + 2x + 1 = -1 + 1

3. Factor the left side as a perfect square: (x + 1)² = 0

4. Solve for x: x + 1 = 0 x = -1

Conclusion

All three methods demonstrate that the quadratic equation x² + 2x + 1 = 0 has a single real root at x = -1. Understanding these different approaches provides you with a versatile toolkit for solving a wide range of quadratic equations. Choosing the most appropriate method depends on the specific equation and your personal preference. This equation, being a perfect square trinomial, lends itself particularly well to factoring, but the quadratic formula and completing the square offer robust alternatives for more complex scenarios.