Which Are The Solutions Of X2 5x 8

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

The quadratic equation x² + 5x + 8 = 0 is a seemingly simple algebraic expression, yet understanding its solution reveals fundamental concepts in mathematics. This article will delve into the various methods of solving this specific equation and explore the broader implications of quadratic equations and their solutions within algebra and beyond.

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 ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0 (if a were 0, it wouldn't be a quadratic equation).

In our specific equation, x² + 5x + 8 = 0, we have a = 1, b = 5, and c = 8. The solutions to this equation, often called roots, represent the values of x that make the equation true. These roots can be real numbers, complex numbers, or a combination of both.

Methods for Solving Quadratic Equations

There are several methods to solve quadratic equations, each with its own advantages and disadvantages. Let's examine the most common approaches and apply them to x² + 5x + 8 = 0.

1. Factoring

Factoring involves expressing the quadratic expression as a product of two linear expressions. This method is the most straightforward when the equation can be easily factored. However, not all quadratic equations are easily factorable. Let's attempt to factor x² + 5x + 8 = 0:

We are looking for two numbers that add up to 5 (the coefficient of x) and multiply to 8 (the constant term). There are no such integers. Therefore, this equation cannot be factored using real numbers. This indicates that the solutions will involve complex numbers.

2. Quadratic Formula

The quadratic formula provides a universal solution for any quadratic equation. It's derived from completing the square and is a powerful tool when factoring fails. The formula is:

x = [-b ± √(b² - 4ac)] / 2a

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

x = [-5 ± √(5² - 4 * 1 * 8)] / 2 * 1

x = [-5 ± √(25 - 32)] / 2

x = [-5 ± √(-7)] / 2

Notice that we have a negative number under the square root. This signifies that the roots are complex numbers, involving the imaginary unit i, where i² = -1.

Therefore, the solutions are:

x = (-5 + i√7) / 2 and x = (-5 - i√7) / 2

These are the exact solutions to the quadratic equation.

3. Completing the Square

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

Let's apply this method to x² + 5x + 8 = 0:

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

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

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

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

5. Solve for x: x = -5/2 ± (i√7)/2

This yields the same solutions as the quadratic formula:

x = (-5 + i√7) / 2 and x = (-5 - i√7) / 2

Understanding the Solutions: Complex Numbers

The solutions we obtained are complex numbers, consisting of a real part and an imaginary part. The imaginary unit, i, allows us to extend the number system beyond real numbers to include solutions for equations like ours that cannot be solved using only real numbers.

The solutions (-5 + i√7) / 2 and (-5 - i√7) / 2 are complex conjugates. Complex conjugates always appear in pairs when solving quadratic equations with real coefficients. This is a significant property of quadratic equations.

Graphical Representation and the Discriminant

A quadratic equation can be graphically represented as a parabola. The roots of the equation represent the x-intercepts of the parabola – the points where the parabola intersects the x-axis.

The discriminant, denoted as Δ (delta), is the expression inside the square root in the quadratic formula: b² - 4ac. The discriminant determines the nature of the roots:

• Δ > 0: Two distinct real roots (parabola intersects the x-axis at two points).
• Δ = 0: One real root (repeated root, parabola touches the x-axis at one point).
• Δ < 0: Two complex conjugate roots (parabola does not intersect the x-axis).

In our equation, x² + 5x + 8 = 0, the discriminant is 5² - 4 * 1 * 8 = -7, which is less than 0. This confirms that the roots are indeed two complex conjugate numbers, as we found using the other methods.

Applications of Quadratic Equations

Quadratic equations are not just abstract mathematical concepts; they have wide-ranging applications in various fields:

• Physics: Calculating projectile motion, analyzing oscillations, and modeling gravitational forces.
• Engineering: Designing structures, analyzing circuits, and optimizing systems.
• Economics: Modeling supply and demand, determining optimal production levels, and analyzing market trends.
• Computer Graphics: Creating curves and shapes, and manipulating objects in 2D and 3D spaces.

Understanding quadratic equations and their solutions is crucial for solving problems in these and many other areas.

Conclusion

Solving the quadratic equation x² + 5x + 8 = 0 demonstrates the importance of various algebraic techniques and highlights the nature of complex numbers. The inability to factor the equation over real numbers leads us to the powerful quadratic formula and the method of completing the square, both of which reveal the complex conjugate roots. The discriminant provides a quick way to determine the nature of the roots without fully solving the equation. This seemingly simple equation underscores the rich mathematical landscape underlying seemingly simple algebraic expressions and their significant applications in diverse fields. The exploration of this equation provides a solid foundation for understanding more complex algebraic concepts and their real-world relevance.