Solve X 2 2x 2 0

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

The equation x² + 2x - 2 = 0 represents a quadratic equation, a fundamental concept in algebra. Solving this equation means finding the values of 'x' that make the equation true. There are several methods to achieve this, each with its own strengths and applications. This article will explore various approaches to solve this specific quadratic equation and will delve into the broader context of solving quadratic equations in general.

Understanding Quadratic Equations

Before we jump into solving x² + 2x - 2 = 0, let's establish a foundational understanding of 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 presence of the x² term distinguishes it from linear equations. The solutions to a quadratic equation are also known as its roots or zeros. A quadratic equation can have up to two real roots, one real root (a repeated root), or two complex roots.

Method 1: Quadratic Formula

The quadratic formula provides a direct and universally applicable method for solving quadratic equations. It's derived from completing the square and offers a concise way to find the roots, regardless of their nature (real or complex). The formula is:

x = (-b ± √(b² - 4ac)) / 2a

In our equation, x² + 2x - 2 = 0, we have a = 1, b = 2, and c = -2. Substituting these values into the quadratic formula, we get:

x = (-2 ± √(2² - 4 * 1 * -2)) / (2 * 1) x = (-2 ± √(4 + 8)) / 2 x = (-2 ± √12) / 2 x = (-2 ± 2√3) / 2 x = -1 ± √3

Therefore, the two solutions are:

• x = -1 + √3
• x = -1 - √3

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

• x ≈ 0.732
• x ≈ -2.732

Method 2: Completing the Square

Completing the square is a powerful technique that transforms the quadratic equation into a perfect square trinomial, making it easier to solve. The steps are as follows:

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

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

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

4. Take the square root of both sides: x + 1 = ±√3

5. Solve for 'x': x = -1 ± √3

This yields the same solutions as the quadratic formula: x = -1 + √3 and x = -1 - √3.

Method 3: Factoring (Not Directly Applicable Here)

Factoring involves expressing the quadratic equation as a product of two linear factors. While this method is efficient for some quadratic equations, it's not directly applicable to x² + 2x - 2 = 0 because this quadratic doesn't factor neatly into integer factors. We would need to resort to more advanced factoring techniques or approximate solutions if attempting this approach.

Understanding the Discriminant

The expression inside the square root in the quadratic formula, b² - 4ac, is known as the discriminant. It provides valuable information about the nature of the roots:

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

In our equation, b² - 4ac = 2² - 4 * 1 * -2 = 12 > 0. This confirms that there are two distinct real roots, as we found using the quadratic formula and completing the square.

Graphical Representation

The equation x² + 2x - 2 = 0 represents a parabola. The x-intercepts of this parabola correspond to the roots of the equation. By plotting the parabola, we can visually confirm the approximate locations of the roots, which we calculated to be approximately 0.732 and -2.732. The parabola opens upwards because the coefficient of x² (which is 1) is positive.

Applications of Quadratic Equations

Quadratic equations appear extensively in various fields:

• Physics: Calculating projectile motion, determining the trajectory of objects under gravity.
• Engineering: Designing structures, analyzing circuits, and modeling systems.
• Economics: Modeling supply and demand curves, optimizing production levels.
• Computer Graphics: Creating curves and shapes in two and three dimensions.
• Mathematics: Solving optimization problems, analyzing functions, and exploring geometric concepts.

Solving Other Quadratic Equations: Examples

Let's illustrate the application of the methods discussed above with additional examples:

Example 1: x² - 4x + 4 = 0

This equation factors easily: (x - 2)(x - 2) = 0, giving a repeated root of x = 2. The discriminant is 0, further confirming this.

Example 2: x² + 4x + 5 = 0

Using the quadratic formula: x = (-4 ± √(16 - 20)) / 2 = (-4 ± √-4) / 2 = -2 ± i. The discriminant is negative, indicating two complex roots.

Example 3: 2x² - 3x - 2 = 0

This equation can be factored as (2x + 1)(x - 2) = 0, resulting in roots x = -1/2 and x = 2.

Example 4: x² + 6x + 9 = 0

Completing the square yields (x + 3)² = 0, therefore x = -3 (repeated root). The discriminant is 0.

Conclusion

Solving quadratic equations is a crucial skill in mathematics and various applied fields. The quadratic formula offers a reliable method for finding solutions, regardless of the nature of the roots. Completing the square provides an alternative approach with insightful geometric interpretations. Factoring, when possible, presents a streamlined approach. Understanding the discriminant offers valuable information about the characteristics of the roots without explicitly calculating them. By mastering these methods, you will have a robust toolset for tackling a wide range of problems involving quadratic equations. Remember to practice regularly and explore different approaches to strengthen your understanding and problem-solving skills.