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Solving the Quadratic Equation: x² + 4x + 1 = 0

The equation x² + 4x + 1 = 0 represents a quadratic equation, a fundamental concept in algebra. Understanding how to solve quadratic equations is crucial for various applications in mathematics, physics, engineering, and computer science. This article will delve into multiple methods for solving this specific equation and explore the broader context of quadratic equations. We'll also discuss the significance of the discriminant and its implications for the nature of the solutions.

Understanding Quadratic Equations

A quadratic equation is an equation of the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The highest power of the variable 'x' is 2, hence the term "quadratic." The solutions to a quadratic equation are also known as its roots or zeros. These roots represent the x-values where the corresponding quadratic function intersects the x-axis.

Our specific equation, x² + 4x + 1 = 0, fits this general form with a = 1, b = 4, and c = 1.

Method 1: Quadratic Formula

The quadratic formula is a universally applicable method for solving any quadratic equation. It provides a direct calculation of the roots based on the coefficients 'a', 'b', and 'c'. The formula is:

x = [-b ± √(b² - 4ac)] / 2a

Let's apply this to our equation:

• a = 1
• b = 4
• c = 1

Substituting these values into the quadratic formula:

x = [-4 ± √(4² - 4 * 1 * 1)] / (2 * 1) x = [-4 ± √(16 - 4)] / 2 x = [-4 ± √12] / 2 x = [-4 ± 2√3] / 2 x = -2 ± √3

Therefore, the two solutions (roots) for the equation x² + 4x + 1 = 0 are:

• x₁ = -2 + √3
• x₂ = -2 - √3

These are the exact solutions. Approximate decimal values can be obtained using a calculator:

• x₁ ≈ -0.268
• x₂ ≈ -3.732

Method 2: Completing the Square

Completing the square is an algebraic technique used to manipulate the quadratic equation into a perfect square trinomial, which can then be easily solved. The steps are:

1. Move the constant term to the right side: x² + 4x = -1

2. Take half of the coefficient of x (which is 4), square it (4/2 = 2, 2² = 4), and add it to both sides: x² + 4x + 4 = -1 + 4 x² + 4x + 4 = 3

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

4. Take the square root of both sides: x + 2 = ±√3

5. Solve for x: x = -2 ± √3

This method yields the same solutions as the quadratic formula: x₁ = -2 + √3 and x₂ = -2 - √3.

Method 3: Factoring (Not Directly Applicable)

Factoring is a method of solving quadratic equations by expressing the quadratic expression as a product of two linear factors. While this method is often efficient, it's not directly applicable to the equation x² + 4x + 1 = 0 because this quadratic expression doesn't factor neatly into rational numbers. We would need to resort to irrational numbers to factor it, making the quadratic formula or completing the square more practical.

The Discriminant: Understanding the Nature of Roots

The expression b² - 4ac within the quadratic formula is called the discriminant. It provides valuable information about the nature of the roots of the quadratic equation:

• If b² - 4ac > 0: The equation has two distinct real roots. This is the case with our equation (16 - 4 = 12 > 0).
• If b² - 4ac = 0: The equation has one real root (a repeated root).
• If b² - 4ac < 0: The equation has two distinct complex roots (roots involving the imaginary unit 'i').

Graphical Representation

The equation x² + 4x + 1 = 0 represents a parabola. The roots of the equation are the x-intercepts of this parabola. Graphing the parabola visually confirms the two real roots we calculated. The parabola intersects the x-axis at approximately x ≈ -0.268 and x ≈ -3.732, aligning with our calculated solutions.

Applications of Quadratic Equations

Quadratic equations have wide-ranging applications in various fields:

• Physics: Calculating projectile motion, determining the trajectory of objects under gravity.
• Engineering: Designing structures, analyzing stresses and strains, modeling oscillations.
• Computer Science: Developing algorithms, solving optimization problems, creating curves in computer graphics.
• Economics: Modeling supply and demand, analyzing market equilibrium.

Solving Related Quadratic Equations

Understanding the methods outlined above allows you to tackle other quadratic equations. For instance, consider the equation 2x² - 6x + 4 = 0. You can apply the quadratic formula, complete the square, or even factor this equation to find its roots. Remember to correctly identify the values of 'a', 'b', and 'c' before applying any method.

Conclusion

Solving the quadratic equation x² + 4x + 1 = 0 illustrates the fundamental principles of quadratic equations. The quadratic formula, completing the square, and even attempting factoring (though not always successful) are valuable tools in your mathematical arsenal. Understanding the discriminant helps predict the nature of the solutions, and the graphical representation provides a visual interpretation. The broad applicability of quadratic equations across numerous fields underscores their importance in mathematics and beyond. Mastering these techniques equips you with the skills to tackle more complex mathematical problems and real-world applications.