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Solving the Quadratic Equation: x² + 2x - 4 = 0

The seemingly simple equation, x² + 2x - 4 = 0, represents a fundamental concept in algebra: solving quadratic equations. While it may look straightforward, understanding the different methods to solve it and the implications of its solution unlocks a deeper understanding of mathematical principles and their applications in various fields. This comprehensive guide will explore multiple approaches to solve this specific quadratic equation, delve into the underlying theory, and discuss the significance of the solutions obtained.

Understanding Quadratic Equations

Before diving into the solution, 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 (in this case, x) is 2. The general form is represented as:

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). In our specific equation, x² + 2x - 4 = 0, we have a = 1, b = 2, and c = -4.

Method 1: Quadratic Formula

The quadratic formula is a powerful tool that provides a direct solution for any quadratic equation. It's derived from completing the square, a technique we'll explore later. The formula is:

x = [-b ± √(b² - 4ac)] / 2a

Plugging in the values from our equation (a = 1, b = 2, c = -4), we get:

x = [-2 ± √(2² - 4 * 1 * -4)] / (2 * 1) x = [-2 ± √(4 + 16)] / 2 x = [-2 ± √20] / 2 x = [-2 ± 2√5] / 2 x = -1 ± √5

Therefore, the two solutions for x are:

• x₁ = -1 + √5
• x₂ = -1 - √5

These are the exact solutions. Approximating them to decimal places gives us:

• x₁ ≈ 1.236
• x₂ ≈ -3.236

Method 2: Completing the Square

Completing the square is a method that manipulates the equation to form a perfect square trinomial, making it easier to solve. Let's apply it to our equation:

x² + 2x - 4 = 0

1. Move the constant term to the right side:

x² + 2x = 4

2. Take half of the coefficient of x (which is 2), square it (1), and add it to both sides:

x² + 2x + 1 = 4 + 1

3. Factor the left side as a perfect square:

(x + 1)² = 5

4. Take the square root of both sides:

x + 1 = ±√5

5. Solve for x:

x = -1 ± √5

This gives us the same solutions as the quadratic formula: x₁ = -1 + √5 and x₂ = -1 - √5.

Method 3: Factoring (Not Directly Applicable)

Factoring involves expressing the quadratic equation as a product of two linear expressions. While this method is efficient for some quadratic equations, it's not directly applicable to x² + 2x - 4 = 0 because it doesn't factor neatly into integers. We would need to use irrational numbers, making the quadratic formula or completing the square more practical in this case.

Understanding the Discriminant (b² - 4ac)

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

• If b² - 4ac > 0: The equation has two distinct real roots (as in our case).
• If b² - 4ac = 0: The equation has one real root (a repeated root).
• If b² - 4ac < 0: The equation has two complex conjugate roots (involving imaginary numbers).

In our equation, b² - 4ac = 20 > 0, indicating two distinct real roots, which we've already found.

Graphical Representation

Graphically, the quadratic equation x² + 2x - 4 = 0 represents a parabola. The solutions (x₁ and x₂) are the x-intercepts of the parabola—the points where the parabola intersects the x-axis. Visualizing this helps understand the meaning of the solutions: they represent the values of x where the function's value is zero.

Applications of Quadratic Equations

Quadratic equations have widespread applications across various fields:

• Physics: Modeling projectile motion, calculating the trajectory of objects under gravity.
• Engineering: Designing structures, analyzing stresses and strains in materials.
• Economics: Optimizing production, determining equilibrium points in markets.
• Computer Graphics: Creating curves and shapes, representing 2D and 3D objects.
• Finance: Calculating compound interest, modeling investment growth.

The ability to solve quadratic equations is crucial for understanding and solving problems within these diverse domains.

Solving Similar Equations

The methods outlined above can be applied to solve a wide range of quadratic equations. The key is to identify the values of a, b, and c and then apply the quadratic formula, complete the square, or attempt factoring (if possible). Remember to always check your solutions by substituting them back into the original equation.

Conclusion

Solving the seemingly simple equation x² + 2x - 4 = 0 offers a profound glimpse into the world of quadratic equations and their significance in mathematics and its applications. By mastering the different methods – the quadratic formula, completing the square, and understanding the concept of the discriminant – we gain the ability to solve a broad spectrum of quadratic equations, opening doors to a deeper appreciation of mathematical principles and their practical relevance. Remember that consistent practice and understanding the underlying concepts are key to becoming proficient in solving quadratic equations and applying this knowledge effectively in various contexts. This foundational understanding lays the groundwork for tackling more complex mathematical challenges in the future.