Which Are The Solutions Of X2 5x 8

Solving the Quadratic Equation: x² + 5x + 8 = 0

This article delves into the solution of the quadratic equation x² + 5x + 8 = 0, exploring various methods and providing a comprehensive understanding of the concepts involved. We will examine the nature of the roots, using the discriminant, and then apply the quadratic formula and other techniques to find the solutions. The article also touches upon the graphical representation of the equation and its connection to the solutions.

Understanding Quadratic Equations

A quadratic equation is a second-degree polynomial equation of the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to this equation, also known as roots or zeros, represent the values of x that satisfy the equation. These roots can be real or complex numbers.

Our specific equation, x² + 5x + 8 = 0, fits this form with a = 1, b = 5, and c = 8.

The Discriminant: Determining the Nature of the Roots

Before jumping into solving the equation, let's analyze the discriminant, denoted by Δ (delta). The discriminant is a crucial part of the quadratic formula and helps determine the nature of the roots:

• Δ = b² - 4ac

For our equation:

Δ = (5)² - 4 * (1) * (8) = 25 - 32 = -7

Since the discriminant is negative (Δ < 0), the quadratic equation has two distinct complex roots. These roots will be complex conjugates of each other, meaning they will have the same real part but opposite imaginary parts.

Solving Using the Quadratic Formula

The quadratic formula provides a general solution for any quadratic equation. It is derived from completing the square and is given by:

• x = (-b ± √Δ) / 2a

Substituting the values from our equation (a = 1, b = 5, c = 8, and Δ = -7):

x = (-5 ± √-7) / 2

Since we have a negative value under the square root, we introduce the imaginary unit 'i', where i² = -1. Therefore, √-7 = √7i.

The solutions are:

• x₁ = (-5 + i√7) / 2
• x₂ = (-5 - i√7) / 2

These are the two complex conjugate roots of the equation x² + 5x + 8 = 0.

Graphical Representation

The graph of a quadratic equation is a parabola. Since our equation has a positive leading coefficient (a = 1), the parabola opens upwards. The fact that the discriminant is negative indicates that the parabola does not intersect the x-axis. This visually confirms that there are no real roots; the roots are complex numbers.

Alternative Methods: Completing the Square

While the quadratic formula is a powerful tool, we can also solve quadratic equations by completing the square. This method involves manipulating the equation to create a perfect square trinomial.

1. Move the constant term to the right side: x² + 5x = -8

2. Take half of the coefficient of x (which is 5/2), square it ((5/2)² = 25/4), and add it to both sides: x² + 5x + 25/4 = -8 + 25/4

3. Simplify the right side: x² + 5x + 25/4 = -7/4

4. Factor the left side as a perfect square: (x + 5/2)² = -7/4

5. Take the square root of both sides: x + 5/2 = ±√(-7/4) = ±(i√7)/2

6. Solve for x: x = -5/2 ± (i√7)/2

This method yields the same complex conjugate roots as the quadratic formula.

Why Complex Roots? Real-World Implications

The occurrence of complex roots might seem abstract, but they have significant applications in various fields, including:

• Electrical Engineering: Complex numbers are fundamental in analyzing alternating current (AC) circuits, where imaginary components represent reactance.
• Quantum Mechanics: Complex numbers are essential in describing wave functions and probabilities in quantum systems.
• Signal Processing: Complex numbers help in analyzing and manipulating signals, including audio and radio waves.
• Fluid Dynamics: Complex analysis is used to solve certain fluid flow problems.

Checking the Solutions

It's always a good practice to verify the solutions. While substituting the complex roots back into the original equation might seem daunting, the process is straightforward: you’ll find that both roots satisfy the equation. Remember to handle the 'i' appropriately using i² = -1.

Conclusion

The quadratic equation x² + 5x + 8 = 0 has two distinct complex conjugate roots: (-5 + i√7) / 2 and (-5 - i√7) / 2. We explored multiple methods to solve this equation, highlighting the importance of the discriminant in determining the nature of the roots and showcasing the application of the quadratic formula and completing the square. The exploration also extends to understanding the graphical interpretation and the significance of complex roots in various fields. This comprehensive approach ensures a thorough understanding of solving quadratic equations, including those with complex solutions. The exploration extends beyond simply finding the answers; it aims to provide a solid grasp of the underlying mathematical principles and their broader applications. By understanding these concepts, you’re not just solving an equation; you're opening doors to a deeper comprehension of mathematics and its role in the world around us.