Solving Quadratic Equations: A Step-by-Step Guide

Hey guys! Let's dive into the fascinating world of quadratic equations. We're going to break down a problem that involves finding the roots of a quadratic equation and then constructing a new equation based on those roots. This is a classic math problem that tests your understanding of quadratic equations, their roots, and how they relate to each other. Don't worry, I'll walk you through it step-by-step, making sure it's super clear and easy to follow. We'll start with the basics, solve the initial equation, and then use that information to find the final answer. So, grab your pencils and let's get started!

Understanding the Problem and Key Concepts

Alright, first things first. We're given a quadratic equation, x^2 + 3x - 10 = 0. The problem tells us that p and q are the roots (also known as solutions) of this equation, and crucially, p - q > 0. This last part is super important because it tells us which root is larger. Remember, the roots of a quadratic equation are the values of x that make the equation true. In other words, if you plug in p or q into the equation, the result will be zero. Our goal is to find a new quadratic equation whose roots are p + 2 and q + 3. This means we need to figure out what the new equation looks like, based on these shifted roots.

So, before we start crunching numbers, let's refresh our memory on some key concepts. A quadratic equation is generally written in the form ax^2 + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. The roots of a quadratic equation can be found using the quadratic formula, but sometimes, like in our case, we can find them by factoring. Factoring is the process of breaking down a quadratic expression into two simpler expressions, which, when multiplied together, give us the original expression. Understanding the relationship between the roots and the coefficients of the quadratic equation is also key. For example, the sum of the roots is -b/a and the product of the roots is c/a. Keep these ideas in mind as we work through the problem.

Now, let's figure out how to solve this. Because we want to find the new quadratic equations, we need to understand how the roots p and q affect it. The given condition is important to help us solve the roots. The roots of a quadratic equation are the values of 'x' that satisfy the equation. In this case, we have a quadratic equation, we'll try to find the roots and then build a new equation from them. This problem combines the knowledge of quadratic equation roots and the process of constructing new equations. You're going to need to recall your knowledge about quadratic formulas, factoring, and the relationship between roots and coefficients. This is the perfect opportunity to boost your understanding of quadratic equations.

Finding the Roots of the Original Equation

Okay, let's get our hands dirty and actually solve for p and q. We have the equation x^2 + 3x - 10 = 0. We can find the roots of this equation by factoring. Factoring involves finding two numbers that multiply to give you the constant term (-10 in this case) and add up to the coefficient of the x term (3 in this case).

Think about it for a second. What two numbers multiply to -10 and add to 3? The numbers 5 and -2 do the trick, right? So, we can rewrite the equation as (x + 5)(x - 2) = 0. Now, for this equation to be true, either (x + 5) = 0 or (x - 2) = 0. Solving these two simple equations gives us our roots. If x + 5 = 0, then x = -5. And if x - 2 = 0, then x = 2.

So, we have two possible roots: -5 and 2. We're given that p - q > 0. This means that p must be the larger root and q must be the smaller root. Therefore, p = 2 and q = -5. Now that we have the values of p and q, we can move on to the next step, which is constructing the new quadratic equation. Remember, it's crucial to correctly identify p and q based on the given condition (p - q > 0). This step highlights the importance of paying attention to detail in mathematical problems. The condition helps us precisely determine the values of p and q. This is also a good reminder that often, there is more than one way to solve a math problem and being able to find the roots allows you to understand how to solve equations.

Remember, if you find that step a little tricky, don't worry! It's perfectly normal. Keep practicing and you will get better. Now we have two roots which is a very important step. Understanding the values of p and q correctly sets you up for the next stage. Knowing p and q is the foundation to constructing the new quadratic equation. This section is all about getting the foundation right. It's like building a house – you need a solid foundation before you can build the walls and the roof. We've got our foundation in place, and we are ready to move on. Let's start the next step!

Constructing the New Quadratic Equation

Alright, we've found our roots! Now the fun begins: constructing the new quadratic equation. We know that the roots of the new equation are p + 2 and q + 3. We already know that p = 2 and q = -5. Let's calculate the new roots.

The first new root is p + 2 = 2 + 2 = 4. The second new root is q + 3 = -5 + 3 = -2.

So, the roots of our new quadratic equation are 4 and -2. Now, how do we use these roots to create an equation? Well, remember that a quadratic equation can be written in the form x^2 - (sum of roots)x + (product of roots) = 0. This is a handy shortcut. It's based on the relationships between the roots and the coefficients of the quadratic equation. Let's use it.

The sum of the new roots is 4 + (-2) = 2. The product of the new roots is 4 * (-2) = -8. Now we can plug these values into our equation: x^2 - (2)x + (-8) = 0. Simplifying this gives us x^2 - 2x - 8 = 0. So, the quadratic equation with roots p + 2 and q + 3 is x^2 - 2x - 8 = 0. And there you have it, guys!

This is a classic example of how to build a new equation based on modified roots. It showcases the versatility of quadratic equations and how their properties can be manipulated. If you ever are stuck on this kind of problem, you should think about roots and their relation to equations. This whole process is more about understanding the relationship between the roots and the coefficients of the quadratic equation. Now you can solve this problem like a pro. This process of constructing the new quadratic equation is a fundamental concept in algebra.

Final Answer and Conclusion

So, based on our calculations, the correct answer is (C) x^2 - 2x - 8 = 0. We've successfully navigated through the problem, finding the roots of the original equation, modifying them, and constructing a new quadratic equation. We used factoring to find the roots, understood the condition p - q > 0, and utilized the relationship between the roots and the coefficients to build the new equation.

Remember that practice makes perfect, and the more you work with quadratic equations, the more comfortable you'll become. Make sure to review the concepts we covered: factoring, the relationship between roots and coefficients, and how to construct a new equation based on modified roots. Keep practicing, and you'll become a quadratic equation master in no time! Also, try to solve similar questions that come in different forms. This is one of those topics in math that can easily give you a headache if you don't keep up with your practice. So, keep studying, guys! And congratulations on getting through this problem. You are doing a great job!