Solving P(x) = 6x² - 5x - 8 A Comprehensive Guide

by Sebastian Müller 50 views

Hey guys! Today, we're diving deep into the fascinating world of quadratic equations, and we're going to dissect the equation P(x) = 6x² - 5x - 8 piece by piece. Whether you're a student grappling with algebra, a math enthusiast, or just someone curious about the mysteries of equations, this guide is for you. We'll break down the equation, explore its properties, and uncover the methods to find its solutions. So, buckle up and let's embark on this mathematical adventure!

Understanding Quadratic Equations: The Basics

Before we jump into the specifics of our equation, let's lay a solid foundation by understanding what quadratic equations are all about. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (in our case, 'x') is 2. The general form of a quadratic equation is ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. If 'a' were zero, the equation would become linear, not quadratic. The coefficients 'a', 'b', and 'c' play crucial roles in determining the shape and position of the parabola that represents the quadratic equation when graphed.

In our equation, P(x) = 6x² - 5x - 8, we can easily identify the coefficients: a = 6, b = -5, and c = -8. These values are the key to unlocking the solutions, also known as roots or zeros, of the equation. The roots are the values of 'x' that make the equation true, meaning when you plug them into the equation, the result is zero. Finding these roots is the primary goal when solving a quadratic equation. Why are these roots so important? Well, they represent the x-intercepts of the parabola, the points where the graph crosses the x-axis. These points are significant in various real-world applications, from physics and engineering to economics and computer science. Imagine modeling the trajectory of a projectile or optimizing a business process – quadratic equations and their roots often hold the key to solving these problems.

Exploring the Discriminant: Unveiling the Nature of Roots

Before we start crunching numbers, let's talk about a powerful tool called the discriminant. The discriminant is a part of the quadratic formula that tells us about the nature of the roots – whether they are real or complex, and whether they are distinct or repeated. It's like a mathematical crystal ball that gives us a sneak peek into the solutions without actually solving the equation. The discriminant is denoted by the Greek letter delta (Δ) and is calculated using the formula: Δ = b² - 4ac. Remember our equation, P(x) = 6x² - 5x - 8? We know a = 6, b = -5, and c = -8. Let's plug these values into the discriminant formula:

Δ = (-5)² - 4 * 6 * (-8) = 25 + 192 = 217

So, the discriminant for our equation is 217. What does this number tell us? The discriminant can be positive, negative, or zero, and each case reveals different information about the roots:

  • If Δ > 0: The equation has two distinct real roots. This means the parabola intersects the x-axis at two different points.
  • If Δ = 0: The equation has one real root (a repeated root). This means the parabola touches the x-axis at exactly one point, the vertex.
  • If Δ < 0: The equation has two complex roots (no real roots). This means the parabola does not intersect the x-axis at all.

In our case, Δ = 217, which is greater than 0. Therefore, we know that the equation P(x) = 6x² - 5x - 8 has two distinct real roots. This is valuable information because it guides our approach to solving the equation. We know we're looking for two real numbers that will make the equation equal to zero. The discriminant has given us a head start in our quest for the roots!

Solving the Quadratic Equation: Methods and Techniques

Now comes the exciting part: actually finding the roots of our equation. There are several methods we can use to solve quadratic equations, each with its own strengths and weaknesses. Let's explore three common methods: factoring, completing the square, and the quadratic formula.

1. Factoring

Factoring is a method that involves breaking down the quadratic expression into two linear factors. It's like reverse multiplication – we're trying to find two expressions that, when multiplied together, give us our original quadratic equation. This method is efficient when the quadratic equation can be easily factored, but it's not always applicable. For example, the equation x² - 4x + 3 can be factored into (x - 1)(x - 3) = 0, which gives us the roots x = 1 and x = 3.

However, our equation, P(x) = 6x² - 5x - 8, is not easily factorable using simple integers. We could try to find two numbers that multiply to 6 * -8 = -48 and add up to -5, but it's not immediately obvious. In such cases, we need to turn to other methods like the quadratic formula or completing the square. Factoring is a powerful tool when it works, but it's not a universal solution for all quadratic equations. It's like having a specialized tool in your toolbox – perfect for the right job, but not suitable for every situation.

2. Completing the Square

Completing the square is a more versatile method that can be used to solve any quadratic equation. It involves manipulating the equation to create a perfect square trinomial on one side, which can then be easily factored. This method is a bit more involved than factoring, but it's a reliable way to find the roots, especially when factoring is difficult or impossible. The process of completing the square involves several steps:

  1. Divide the equation by the coefficient of x² (if it's not 1). In our case, we would divide the entire equation by 6: x² - (5/6)x - (8/6) = 0.
  2. Move the constant term to the right side of the equation: x² - (5/6)x = 4/3.
  3. Take half of the coefficient of the x term, square it, and add it to both sides of the equation. The coefficient of x is -5/6, so half of it is -5/12, and squaring it gives us 25/144. Adding this to both sides, we get: x² - (5/6)x + 25/144 = 4/3 + 25/144.
  4. Rewrite the left side as a perfect square trinomial: (x - 5/12)² = 4/3 + 25/144.
  5. Simplify the right side: (x - 5/12)² = 217/144.
  6. Take the square root of both sides: x - 5/12 = ±√(217/144).
  7. Solve for x: x = 5/12 ± √217 / 12.

This gives us two roots: x = (5 + √217) / 12 and x = (5 - √217) / 12. While completing the square is a powerful method, it can be a bit tedious, especially when dealing with fractions. This is where the quadratic formula comes in handy – it's a streamlined way to achieve the same result.

3. The Quadratic Formula

The quadratic formula is the ultimate tool for solving quadratic equations. It's a formula that directly gives you the roots of any quadratic equation in the form ax² + bx + c = 0. It's like a magic wand that instantly reveals the solutions, no matter how complex the equation may seem. The quadratic formula is given by:

x = (-b ± √(b² - 4ac)) / 2a

Notice the familiar discriminant (b² - 4ac) lurking under the square root! This is no coincidence – the discriminant is an integral part of the formula. Let's apply the quadratic formula to our equation, P(x) = 6x² - 5x - 8. We have a = 6, b = -5, and c = -8. Plugging these values into the formula, we get:

x = (5 ± √((-5)² - 4 * 6 * (-8))) / (2 * 6)

x = (5 ± √(25 + 192)) / 12

x = (5 ± √217) / 12

This gives us the same roots we found using completing the square: x = (5 + √217) / 12 and x = (5 - √217) / 12. The quadratic formula is a reliable and efficient method for solving quadratic equations, and it's often the preferred choice, especially when dealing with complex coefficients or equations that are difficult to factor.

The Solutions: Unveiling the Roots of P(x) = 6x² - 5x - 8

We've reached the final destination! We've explored the equation P(x) = 6x² - 5x - 8 using various methods, and we've arrived at the solutions. The roots of the equation are:

x₁ = (5 + √217) / 12 x₂ = (5 - √217) / 12

These are the two real values of 'x' that make the equation equal to zero. They represent the points where the parabola intersects the x-axis. We can approximate these values using a calculator:

x₁ ≈ 1.628 x₂ ≈ -0.795

So, the parabola intersects the x-axis at approximately x = 1.628 and x = -0.795. These roots provide valuable information about the behavior of the quadratic function and can be used in various applications. For example, if this equation represented the profit of a business, the roots would represent the break-even points where the profit is zero. Understanding the roots of a quadratic equation is like understanding the heartbeat of a mathematical system – it gives you insights into its behavior and its potential.

Real-World Applications: Why Quadratic Equations Matter

Quadratic equations are not just abstract mathematical concepts; they have a wide range of applications in the real world. They pop up in various fields, from physics and engineering to economics and computer science. Let's explore some exciting examples of how quadratic equations make a difference:

  • Physics: Imagine throwing a ball into the air. The path the ball takes, its trajectory, can be modeled using a quadratic equation. The equation takes into account the initial velocity, the angle of launch, and the force of gravity. By solving the quadratic equation, we can determine the maximum height the ball will reach and how far it will travel before hitting the ground. This is crucial in sports like baseball and basketball, where understanding trajectory is essential.
  • Engineering: Engineers use quadratic equations to design bridges, buildings, and other structures. For example, the cables of a suspension bridge form a parabolic shape, which can be described by a quadratic equation. By carefully analyzing the equation, engineers can ensure the structural integrity and stability of the bridge. Similarly, quadratic equations are used in the design of antennas, lenses, and other optical devices.
  • Economics: Quadratic equations can be used to model supply and demand curves in economics. The point where the supply and demand curves intersect represents the equilibrium price, which is a crucial concept in market analysis. Businesses can use quadratic equations to optimize their pricing strategies and maximize profits. For example, a company might use a quadratic equation to model the relationship between the price of a product and the quantity sold, and then find the price that yields the highest revenue.
  • Computer Science: Quadratic equations are used in computer graphics to create smooth curves and surfaces. They are also used in optimization algorithms, which are used to solve problems like finding the shortest path between two points or allocating resources efficiently. In machine learning, quadratic equations are used in support vector machines (SVMs), which are powerful algorithms for classification and regression.

These are just a few examples of the many ways quadratic equations are used in the real world. From the trajectory of a ball to the design of a bridge, quadratic equations are essential tools for solving a wide range of problems. Understanding quadratic equations is not just about mastering algebra; it's about unlocking the power to understand and shape the world around us.

Conclusion: The Power of Quadratic Equations

We've journeyed through the world of quadratic equations, dissected the equation P(x) = 6x² - 5x - 8, and uncovered its secrets. We've learned about the discriminant, explored various methods for solving quadratic equations, and discovered the real-world applications of these powerful mathematical tools. Quadratic equations are not just abstract formulas; they are the language of curves, trajectories, and optimization. They are the key to understanding and solving problems in physics, engineering, economics, computer science, and many other fields.

So, the next time you encounter a quadratic equation, don't be intimidated. Remember the tools and techniques we've discussed, and embrace the challenge. You now have the knowledge and skills to decode these equations and unlock their hidden potential. Keep exploring, keep learning, and keep applying the power of quadratic equations to make a difference in the world. You've got this!