Solving Quadratic Equations: Find Roots Of P^2 - P - 132 = 0
Are you grappling with quadratic equations and trying to figure out how to find their roots? You're not alone! Quadratic equations might seem intimidating at first, but with a systematic approach, they can be solved quite easily. In this article, we'll dive deep into the process of solving the quadratic equation p^2 - p - 132 = 0. We'll break down each step, making it super clear and easy to follow. By the end, you'll not only know the answer but also understand the why behind it. So, let's put on our math hats and get started!
Understanding Quadratic Equations
Before we jump into solving our specific equation, let's quickly recap what a quadratic equation actually is. In simple terms, a quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (in our case, 'p') is 2. The general form of a quadratic equation is ax^2 + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'x' is the variable. Recognizing this form is crucial because it sets the stage for applying different solution methods.
Now, why are we so interested in finding the 'roots' of these equations? Well, the roots, also known as solutions or zeros, are the values of the variable that make the equation true. Graphically, these roots represent the points where the parabola (the graph of a quadratic equation) intersects the x-axis. Finding these roots has wide applications in various fields, from physics and engineering to economics and computer science. So, mastering this skill is definitely worth the effort!
Methods to Solve Quadratic Equations
There are several methods to solve quadratic equations, each with its own strengths and best-use cases. The most common methods include:
- Factoring: This method involves breaking down the quadratic expression into the product of two linear factors. It's often the quickest method when applicable, but it requires a bit of intuition and pattern recognition.
- Completing the Square: This method involves manipulating the equation to create a perfect square trinomial on one side. It's a powerful method that can solve any quadratic equation, but it can be a bit more involved than factoring.
- Quadratic Formula: This is a universal formula that can solve any quadratic equation, regardless of its complexity. It's a reliable method, especially when factoring or completing the square becomes cumbersome.
For our equation, p^2 - p - 132 = 0, we'll explore the factoring method first, as it often provides the most straightforward solution if the equation is factorable. If factoring proves difficult, we can always resort to the quadratic formula. Let's dive into factoring!
Solving p^2 - p - 132 = 0 by Factoring
The factoring method hinges on the idea of reversing the distributive property. We're essentially trying to find two binomials that, when multiplied together, give us our original quadratic expression. The general approach involves finding two numbers that add up to the coefficient of the linear term (the 'b' in ax^2 + bx + c = 0) and multiply to the constant term (the 'c').
In our equation, p^2 - p - 132 = 0, the coefficient of the linear term (p) is -1, and the constant term is -132. So, we need to find two numbers that add up to -1 and multiply to -132. This is where a little bit of number sense and trial-and-error comes into play. We start by thinking about factors of 132. After some thought, we might realize that 11 and 12 are factors of 132 (since 11 * 12 = 132). Now, we need to adjust the signs to meet our conditions. If we make 12 negative and keep 11 positive, we have -12 * 11 = -132 and -12 + 11 = -1. Bingo! We've found our numbers.
Now that we have our numbers, -12 and 11, we can rewrite the quadratic equation in its factored form. The factored form will look like this: (p + m)(p + n) = 0, where 'm' and 'n' are the numbers we found. In our case, this translates to (p - 12)(p + 11) = 0. This is a crucial step because it transforms our quadratic equation into a product of two factors, making it much easier to solve.
Finding the Roots
Now that we have the equation in factored form, (p - 12)(p + 11) = 0, we can use the zero-product property to find the roots. The zero-product property states that if the product of two factors is zero, then at least one of the factors must be zero. In other words, if A * B = 0, then either A = 0 or B = 0 (or both).
Applying this property to our equation, we set each factor equal to zero and solve for 'p'. This gives us two simple linear equations:
- p - 12 = 0
- p + 11 = 0
Solving the first equation, we add 12 to both sides, which gives us p = 12. Solving the second equation, we subtract 11 from both sides, which gives us p = -11. And there you have it! We've found the roots of the equation p^2 - p - 132 = 0.
Verifying the Solution
It's always a good practice to verify your solution to make sure you haven't made any mistakes along the way. We can do this by plugging the roots we found back into the original equation and checking if they satisfy the equation. Let's start with p = 12:
(12)^2 - 12 - 132 = 144 - 12 - 132 = 0
The equation holds true for p = 12. Now let's check p = -11:
(-11)^2 - (-11) - 132 = 121 + 11 - 132 = 0
The equation also holds true for p = -11. This confirms that our roots, p = 12 and p = -11, are indeed the solutions to the equation p^2 - p - 132 = 0.
The Answer and Its Significance
So, the roots of the equation p^2 - p - 132 = 0 are p = 12 and p = -11. This means that if we substitute either 12 or -11 for 'p' in the equation, the equation will be true. These roots represent the x-intercepts of the parabola defined by the quadratic equation. Understanding the roots of a quadratic equation is fundamental in many mathematical and real-world applications.
Think about it this way: if you were modeling the trajectory of a projectile (like a ball thrown in the air), the roots of the quadratic equation representing the trajectory would tell you where the projectile hits the ground. Similarly, in business and economics, quadratic equations can model profit curves, and the roots can represent break-even points. So, this seemingly simple math skill has powerful implications!
Alternative Methods: Quadratic Formula
While we successfully solved our equation by factoring, it's worth briefly mentioning the quadratic formula as an alternative method. The quadratic formula is a powerful tool that can solve any quadratic equation, even those that are difficult or impossible to factor. The formula is:
p = [-b ± √(b^2 - 4ac)] / 2a
Where 'a', 'b', and 'c' are the coefficients of the quadratic equation in the general form ax^2 + bx + c = 0. In our equation, p^2 - p - 132 = 0, we have a = 1, b = -1, and c = -132. If we plug these values into the quadratic formula, we'll arrive at the same solutions: p = 12 and p = -11.
The quadratic formula is a bit more computationally intensive than factoring, but it's a reliable method that always works. It's a great tool to have in your mathematical arsenal.
Conclusion: Mastering Quadratic Equations
Congratulations! You've successfully navigated the process of solving the quadratic equation p^2 - p - 132 = 0. We've covered the basics of quadratic equations, explored the factoring method, found the roots, and even touched upon the quadratic formula as an alternative approach. Remember, practice makes perfect, so don't hesitate to tackle more quadratic equations to solidify your understanding.
Finding the roots of quadratic equations is a fundamental skill in mathematics with far-reaching applications. By mastering this skill, you're equipping yourself with a powerful tool for solving a wide range of problems. Whether you're a student tackling algebra, an engineer designing structures, or an economist modeling market trends, understanding quadratic equations will undoubtedly come in handy. So, keep practicing, keep exploring, and keep expanding your mathematical horizons!
For further reading and more examples on solving quadratic equations, you can visit Khan Academy's Quadratic Equations Section. This resource provides a wealth of information and practice problems to help you master this important topic.
