Solving Quadratic Equations: A Step-by-Step Guide

Hey guys! Let's dive into the world of quadratic equations and tackle a problem that seems a bit intimidating at first glance. We're going to break down the question, understand the concepts, and find a solution that's easy to follow. So, let's get started!

Understanding the Problem: Roots and Quadratic Equations

Alright, so the first thing we need to do is understand what the question is asking. We are given a quadratic equation, x² - 3x + 5 = 0, and we know that α and β are its roots. Remember, the roots of a quadratic equation are the values of x that satisfy the equation, meaning when you plug them back into the equation, it equals zero. The question wants us to find the value of α² + β² - 3αβ. This involves knowing a little bit about the properties of quadratic equations and how their roots behave.

To solve this, we're going to use some important relationships that always hold true for quadratic equations. Let's refresh our memory: For a quadratic equation in the standard form ax² + bx + c = 0, the sum of the roots (α + β) is equal to -b/a, and the product of the roots (αβ) is equal to c/a. These are super useful tools, and we'll be using them to solve this problem.

Now, let's break down the specific equation we're working with: x² - 3x + 5 = 0. In this case, a = 1, b = -3, and c = 5. Using the relationships mentioned above, we can find the sum and the product of the roots of this equation. Keep in mind that understanding these fundamental concepts is like having the secret key to unlock a whole bunch of math problems, so it's worth taking the time to make sure you get it.

Step 1: Find the Sum and Product of the Roots

Let's start by calculating the sum of the roots (α + β). Using the formula -b/a, we get: α + β = -(-3)/1 = 3. So the sum of the roots is 3. Now, let's calculate the product of the roots (αβ). Using the formula c/a, we get: αβ = 5/1 = 5. Great! We've got both the sum and the product.

Now that we have the sum and product of the roots, we are equipped with what we need to figure out what is the result of the equation. From here, we will need to do a bit of manipulation of the given equation in order to calculate the result.

Step 2: Manipulate the Expression

Okay, now we want to find the value of α² + β² - 3αβ. We already know αβ = 5, so we're part of the way there. The term α² + β² can be a bit tricky to calculate directly, but we can rewrite it using the square of the sum of the roots. Remember the algebraic identity (α + β)² = α² + 2αβ + β². We can rearrange this to get α² + β² = (α + β)² - 2αβ. Do you see what we did here? We've managed to express α² + β² in terms of quantities we already know: the sum and product of the roots. This is pretty cool, right?

So, let's plug in the values we found earlier. α + β = 3, so (α + β)² = 3² = 9. Therefore, α² + β² = 9 - 2αβ. We know αβ = 5, so α² + β² = 9 - 2(5) = 9 - 10 = -1. Now, we just need to substitute that into the equation.

Step 3: Calculate the Final Value

Finally, we can calculate the value of the original expression α² + β² - 3αβ. We found that α² + β² = -1 and we know that αβ = 5. So, substituting these values, we get: -1 - 3(5) = -1 - 15 = -16*. Therefore, the answer to the question is -16. We did it, guys!

So to sum it up, we first found the sum and product of the roots using the standard formula for quadratic equations. After, we manipulated the equation given to us using algebraic identities. Finally, we plugged in the variables into the equation to arrive at the final result.

The Answer and Why It Matters

The correct answer is B. -16. Knowing how to solve this type of problem is crucial because it reinforces your understanding of quadratic equations and algebraic manipulation, which are fundamental in many areas of mathematics and science. It helps you to think critically and solve complex problems in a systematic manner.

And there you have it! We've successfully solved the problem. Congrats on making it this far, and I hope this explanation makes the process easier to follow. Keep practicing, and you'll become a pro at solving quadratic equations in no time. See ya!

Additional Tips for Tackling Quadratic Equations

Practice Makes Perfect: The more you practice, the more comfortable you'll become with these types of problems. Work through different examples and try to understand the reasoning behind each step.

Know Your Formulas: Make sure you memorize the formulas for the sum and product of roots (-b/a and c/a, respectively), and be familiar with algebraic identities like (α + β)² = α² + 2αβ + β².

Break It Down: Don't try to solve everything at once. Break the problem down into smaller, manageable steps. This makes it easier to understand and reduces the chance of making mistakes.

Double-Check Your Work: Always double-check your calculations, especially when dealing with negative numbers and algebraic manipulations. A small mistake can lead to a completely wrong answer.

Seek Help When Needed: If you get stuck, don't hesitate to ask for help from your teacher, a tutor, or a study group. Sometimes, a fresh perspective can make all the difference. You've got this!

Problem 6: Working with Roots of Equations

Let's move on to another type of problem often encountered when working with quadratic equations. This time, we're going to discuss the concept of finding the roots of an equation. We will talk about the conditions and methods of finding it.

Introduction to the Roots of an Equation

In mathematics, the roots of an equation refer to the values of the variables that satisfy the equation. In the context of quadratic equations, the roots are the values of x that make the equation ax² + bx + c = 0 true. These roots are also known as the solutions or zeros of the equation. Understanding how to find the roots is essential for solving the equation.

There are several methods to find the roots of an equation, depending on the nature of the equation and the desired level of accuracy. Some common methods include factoring, completing the square, and using the quadratic formula. For example, if you are solving for an equation that has an easier factored answer, it might be useful to factor the equation. However, if the factoring is too complex, we can use the quadratic formula to get the answer.

Methods for Finding Roots

Factoring: Factoring involves breaking down the quadratic expression into two simpler expressions, typically in the form of (x - p)(x - q) = 0. If we can factor the quadratic equation, the roots are simply the values of x that make each factor equal to zero. For instance, if the equation can be factored into (x - 2)(x - 3) = 0, then the roots are x = 2 and x = 3.

Completing the Square: Completing the square is a method used to rewrite the quadratic equation in a form that can be easily solved for x. This method involves manipulating the equation to create a perfect square trinomial on one side. While this method can be useful, it's often more complex than other methods.

Quadratic Formula: The quadratic formula is a general formula that provides the roots of a quadratic equation for any values of a, b, and c. The formula is given by x = (-b ± √(b² - 4ac)) / 2a. This formula is a powerful tool because it can be used to find the roots of any quadratic equation, regardless of whether it can be easily factored or not.

Graphical Methods: Another way to find the roots of an equation is to graph the function and find the points where it intersects the x-axis. These points represent the values of x for which y = 0, thus the roots of the equation. These methods are often visual, so they can be helpful for getting an overall picture of how the equation looks.

Properties of Roots

The nature of the roots, whether they are real, complex, or repeated, depends on the discriminant (b² - 4ac). The discriminant tells us about the number and type of roots the equation has.

If b² - 4ac > 0: The equation has two distinct real roots.

If b² - 4ac = 0: The equation has one real root (a repeated root).

If b² - 4ac < 0: The equation has two complex conjugate roots.

Understanding the nature of the roots can help you determine whether the equation has a solution, and if so, what type of solution it is. Therefore, it is important to know the properties of each discriminant.

Conclusion

Mastering the concepts related to roots of equations is crucial for success in mathematics. By understanding the definition, methods, and properties of roots, you can tackle a wide range of mathematical problems and deepen your understanding of algebra. Keep practicing and using these methods, and you'll become more confident in solving quadratic equations.