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 fundamental problem in algebra. While seemingly simple, understanding its solution reveals key concepts in quadratic equations and their applications. This comprehensive guide will explore various methods to solve this equation, delve into the nature of its solutions, and discuss the broader implications within mathematics and other fields.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (x in this case) is 2. The general form is ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. Our specific equation, x² + 5x + 8 = 0, fits this form with a = 1, b = 5, and c = 8.

The solutions to a quadratic equation are also known as its roots or zeros. These represent the x-values that make the equation true. Quadratic equations can have two real solutions, one real solution (a repeated root), or two complex solutions. The type of solution depends on the discriminant (more on that later).

Method 1: Using the Quadratic Formula

The quadratic formula provides a direct and universally applicable method for solving any quadratic equation. The formula is derived from completing the square and is given by:

x = [-b ± √(b² - 4ac)] / 2a

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

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 indicates that the solutions are complex numbers. Complex numbers involve the imaginary unit 'i', where i² = -1. Therefore, the solutions are:

x = [-5 ± i√7] / 2

This gives us two distinct complex solutions:

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

These solutions are complex conjugates of each other, meaning they have the same real part (-5/2) but opposite imaginary parts.

Method 2: Completing the Square

Completing the square is a technique that transforms the quadratic equation into a perfect square trinomial, making it easier to solve. The process involves manipulating the equation to create a square of a binomial.

1. Move the constant term: Subtract 8 from both sides of the equation: x² + 5x = -8

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

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

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

5. Isolate x: Subtract 5/2 from both sides: x = -5/2 ± (i√7)/2

This yields the same complex solutions as the quadratic formula.

Method 3: Graphical Representation

While the previous methods provided algebraic solutions, visualizing the equation graphically helps understand the nature of its roots. The graph of a quadratic equation is a parabola. The x-intercepts of the parabola represent the real roots of the equation. Since our equation has complex roots, the parabola will not intersect the x-axis.

To graph the equation, we can find the vertex and the y-intercept. The x-coordinate of the vertex is given by -b/2a = -5/2. Substituting this into the equation gives the y-coordinate: (-5/2)² + 5(-5/2) + 8 = -7/4. So, the vertex is at (-5/2, -7/4). The y-intercept is found by setting x = 0, which gives y = 8.

Plotting these points and sketching the parabola shows that it opens upwards (since a > 0) and lies entirely above the x-axis, confirming the absence of real roots.

The Discriminant: Determining the Nature of Roots

The discriminant (Δ) of a quadratic equation is the expression under the square root in the quadratic formula: b² - 4ac. The discriminant determines the nature of the roots:

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

In our equation, x² + 5x + 8 = 0, the discriminant is:

Δ = 5² - 4 * 1 * 8 = 25 - 32 = -7

Since Δ < 0, we confirm that the equation has two complex conjugate roots, as we found using the quadratic formula and completing the square.

Applications of Quadratic Equations

Quadratic equations have widespread applications in various fields:

• Physics: Projectile motion, calculating the trajectory of objects under gravity.
• Engineering: Designing structures, analyzing stresses and strains.
• Economics: Modeling supply and demand curves, optimizing profit.
• Computer graphics: Creating curves and shapes.
• Signal processing: Analyzing and manipulating signals.

Conclusion: Understanding the Complex Solutions

Solving x² + 5x + 8 = 0 demonstrates the importance of understanding different solution methods for quadratic equations, particularly when dealing with complex roots. The quadratic formula, completing the square, and graphical representation all offer valuable perspectives. The discriminant provides a quick way to determine the nature of the roots without explicitly solving the equation. While the solutions are complex numbers, they are equally valid and hold significance within the broader context of mathematics and its applications. The understanding gained extends beyond simple equation solving and provides a foundation for more advanced mathematical concepts. The equation, though seemingly simple, opens a door to a deeper understanding of quadratic equations and their diverse applications in various fields.