X 2 2x 1 0

Decoding the Mystery: A Deep Dive into x² + 2x + 1 = 0

The seemingly simple quadratic equation, x² + 2x + 1 = 0, holds a wealth of mathematical concepts within its concise form. This article will explore this equation in detail, unraveling its solutions, demonstrating different solution methods, examining its graphical representation, and exploring its broader significance in algebra and beyond. Understanding this equation provides a strong foundation for tackling more complex quadratic problems and grasping fundamental algebraic principles.

Introduction: Understanding Quadratic Equations

Before delving into the specifics of x² + 2x + 1 = 0, let's establish a foundational understanding of quadratic equations. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (usually 'x') is 2. The general form of a quadratic equation is ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero (otherwise, it wouldn't be a quadratic equation). Our target equation, x² + 2x + 1 = 0, fits this general form perfectly, with a = 1, b = 2, and c = 1.

Method 1: Factoring the Quadratic Expression

One of the most straightforward ways to solve x² + 2x + 1 = 0 is through factoring. Factoring involves expressing the quadratic expression as a product of two linear expressions. Notice that x² + 2x + 1 is a perfect square trinomial. This means it can be factored into the square of a binomial.

Observe the following:

(x + 1)(x + 1) = x² + x + x + 1 = x² + 2x + 1

Therefore, our equation can be rewritten as:

(x + 1)² = 0

To solve for 'x', we take the square root of both sides:

√(x + 1)² = √0

x + 1 = 0

Subtracting 1 from both sides gives us the solution:

x = -1

This indicates that the equation has a single, repeated root, or a root of multiplicity 2. This means the parabola represented by the equation touches the x-axis at only one point, x = -1.

Method 2: Using the Quadratic Formula

The quadratic formula is a powerful tool for solving any quadratic equation, regardless of whether it can be easily factored. The formula is derived from completing the square and provides a direct route to the solution(s). The general quadratic formula is:

x = [-b ± √(b² - 4ac)] / 2a

Substituting the values from our equation (a = 1, b = 2, c = 1) into the quadratic formula yields:

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

x = [-2 ± √(4 - 4)] / 2

x = [-2 ± √0] / 2

x = -2 / 2

x = -1

Again, we arrive at the solution x = -1, confirming our previous result obtained through factoring.

Method 3: Completing the Square

Completing the square is another algebraic technique used to solve quadratic equations. This method involves manipulating the equation to create a perfect square trinomial, which can then be factored easily.

Starting with x² + 2x + 1 = 0:

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

2. Take half of the coefficient of 'x' (which is 2), square it (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' as we did in the factoring method: x = -1

Graphical Representation: Understanding the Parabola

The equation x² + 2x + 1 = 0 represents a parabola. A parabola is a U-shaped curve that is symmetrical about its vertex (the highest or lowest point on the curve). The parabola representing our equation opens upwards because the coefficient of x² (which is 1) is positive. The fact that the equation has only one solution means that the parabola touches the x-axis at only one point, the vertex itself, located at x = -1 and y = 0.

The Significance of Repeated Roots

The fact that x² + 2x + 1 = 0 has a repeated root (x = -1) has important implications. It indicates that the discriminant (the expression inside the square root in the quadratic formula, b² - 4ac) is equal to zero. The discriminant determines the nature of the roots of a quadratic equation:

• b² - 4ac > 0: Two distinct real roots.
• b² - 4ac = 0: One repeated real root (as in our case).
• b² - 4ac < 0: Two complex conjugate roots (involving imaginary numbers).

In our case, the discriminant is 0, indicating a parabola that is tangent to the x-axis, hence the repeated root.

Extending the Concept: Applications in Calculus and Beyond

The understanding of quadratic equations, and specifically the concept of repeated roots, extends beyond basic algebra. In calculus, repeated roots are crucial in determining the multiplicity of roots of a polynomial, influencing the behavior of the function near those roots. For instance, the presence of a repeated root can signify a point of inflection or a local minimum/maximum, depending on the context.

Frequently Asked Questions (FAQ)

Q: What is a perfect square trinomial?

A: A perfect square trinomial is a trinomial (a three-term polynomial) that can be factored into the square of a binomial. For example, x² + 2x + 1 is a perfect square trinomial because it factors into (x + 1)².

Q: Can all quadratic equations be solved by factoring?

A: No, not all quadratic equations can be easily factored. The quadratic formula is a more general method that can solve any quadratic equation, regardless of its factorability.

Q: What does it mean when the discriminant is negative?

A: A negative discriminant indicates that the quadratic equation has no real roots. The roots are complex numbers, involving the imaginary unit 'i' (where i² = -1).

Q: How does the graph of a quadratic equation with a repeated root differ from one with two distinct roots?

A: A quadratic equation with two distinct roots intersects the x-axis at two different points. A quadratic equation with a repeated root (like ours) touches the x-axis at only one point, which is the vertex of the parabola.

Conclusion: Mastering the Fundamentals

The equation x² + 2x + 1 = 0, although seemingly simple, serves as a powerful illustration of several key concepts in algebra. Through factoring, the quadratic formula, and completing the square, we've demonstrated multiple pathways to arrive at the solution, x = -1. Understanding the graphical representation of the equation as a parabola with a repeated root and grasping the significance of the discriminant reinforces the fundamental principles that underpin more advanced mathematical concepts. The journey through this seemingly simple equation underscores the importance of a solid foundation in algebra and its relevance in broader mathematical contexts. Mastering these foundational concepts opens the door to tackling more complex mathematical challenges with confidence and competence.