Finding The Roots Of Y=x²-3x-10 A Step-by-Step Guide

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Hey there, math enthusiasts! Ever found yourself staring at a quadratic equation, wondering how to decipher its secrets? Well, you're in the right place! Today, we're diving deep into the fascinating world of quadratic equations, specifically focusing on finding the roots of the equation y = x² - 3x - 10. We'll break down the concepts, explore different methods, and guide you through the process step-by-step. So, buckle up and get ready to conquer those quadratic equations!

Understanding Quadratic Equations and Their Roots

Before we jump into solving our equation, let's take a moment to understand what quadratic equations are and what we mean by their "roots." 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 0. Our equation, y = x² - 3x - 10, fits this form perfectly, with a = 1, b = -3, and c = -10.

Now, what are roots? Simply put, the roots of a quadratic equation are the values of x that make the equation true, or in other words, the values of x that make y equal to 0. Graphically, the roots are the points where the parabola represented by the quadratic equation intersects the x-axis. Finding these roots is a fundamental skill in algebra and has numerous applications in various fields, from physics and engineering to economics and computer science. To find the roots, we essentially need to solve the equation x² - 3x - 10 = 0. There are several methods to do this, and we'll explore the most common ones in detail.

Methods for Finding the Roots of Quadratic Equations

There are primarily three methods for finding the roots of a quadratic equation: factoring, using the quadratic formula, and completing the square. Each method has its strengths and weaknesses, and the best approach often depends on the specific equation you're dealing with. Let's take a closer look at each of these methods.

1. Factoring: The Art of Decomposition

Factoring is a powerful technique that involves expressing the quadratic expression as a product of two linear factors. This method is particularly efficient when the quadratic expression can be factored easily. The key idea behind factoring is to reverse the process of expanding two binomials. For example, if we have (x + p)(x + q), expanding it gives us x² + (p + q)x + pq. So, when factoring, we need to find two numbers, p and q, such that their sum is equal to the coefficient of the x term (b) and their product is equal to the constant term (c). In our equation, x² - 3x - 10 = 0, we need to find two numbers that add up to -3 and multiply to -10. A little bit of thinking reveals that these numbers are -5 and 2, since (-5) + 2 = -3 and (-5) * 2 = -10. Therefore, we can factor the equation as (x - 5)(x + 2) = 0. Now, for the product of two factors to be zero, at least one of them must be zero. This gives us two separate equations: x - 5 = 0 and x + 2 = 0. Solving these equations, we get x = 5 and x = -2. Thus, the roots of the equation x² - 3x - 10 = 0 are 5 and -2. Factoring is a great method when the roots are integers and the factoring is relatively straightforward. However, it can be challenging when the roots are not integers or when the factoring is not immediately obvious.

2. The Quadratic Formula: A Universal Solution

The quadratic formula is a foolproof method that works for any quadratic equation, regardless of whether it can be factored easily or not. It provides a direct solution for the roots in terms of the coefficients a, b, and c. The formula is derived by completing the square on the general quadratic equation ax² + bx + c = 0, and it is given by:

x = (-b ± √(b² - 4ac)) / 2a

This formula might look a bit intimidating at first, but it's actually quite straightforward to use. Simply plug in the values of a, b, and c from your equation, and you'll get the roots. The "±" sign indicates that there are two possible solutions, one with the plus sign and one with the minus sign. In our case, a = 1, b = -3, and c = -10. Substituting these values into the quadratic formula, we get:

x = (3 ± √((-3)² - 4 * 1 * -10)) / (2 * 1)

x = (3 ± √(9 + 40)) / 2

x = (3 ± √49) / 2

x = (3 ± 7) / 2

So, the two solutions are:

x₁ = (3 + 7) / 2 = 10 / 2 = 5

x₂ = (3 - 7) / 2 = -4 / 2 = -2

As you can see, we arrived at the same roots (5 and -2) as we did with factoring. The quadratic formula is a reliable method that always works, but it can be a bit more computationally intensive than factoring, especially if the coefficients are large or involve fractions. However, it's an invaluable tool to have in your mathematical arsenal.

3. Completing the Square: Transforming the Equation

Completing the square is another method for solving quadratic equations that involves manipulating the equation to create a perfect square trinomial on one side. A perfect square trinomial is a trinomial that can be factored as (x + p)² or (x - p)². This method is particularly useful for understanding the derivation of the quadratic formula and for solving certain types of quadratic equations. To complete the square for the equation x² - 3x - 10 = 0, we first move the constant term to the right side of the equation:

x² - 3x = 10

Next, we take half of the coefficient of the x term (-3), square it ((-3/2)² = 9/4), and add it to both sides of the equation. This is the crucial step in creating the perfect square trinomial:

x² - 3x + 9/4 = 10 + 9/4

Now, the left side of the equation is a perfect square trinomial, which can be factored as (x - 3/2)². The right side simplifies to 49/4:

(x - 3/2)² = 49/4

Taking the square root of both sides, we get:

x - 3/2 = ±√(49/4)

x - 3/2 = ±7/2

Adding 3/2 to both sides, we get:

x = 3/2 ± 7/2

So, the two solutions are:

x₁ = 3/2 + 7/2 = 10/2 = 5

x₂ = 3/2 - 7/2 = -4/2 = -2

Again, we find the same roots (5 and -2). Completing the square can be a bit more involved than factoring or using the quadratic formula, but it provides a deeper understanding of the structure of quadratic equations and is a valuable technique to master.

Applying the Methods to Our Equation: y=x²-3x-10

Now that we've explored the different methods for finding the roots of quadratic equations, let's apply them to our specific equation, y = x² - 3x - 10. As we've already demonstrated in the previous sections, we can find the roots using factoring, the quadratic formula, and completing the square. Let's recap the solutions:

  • Factoring: We factored the equation as (x - 5)(x + 2) = 0, which gave us the roots x = 5 and x = -2.
  • Quadratic Formula: We used the formula x = (-b ± √(b² - 4ac)) / 2a with a = 1, b = -3, and c = -10, which also gave us the roots x = 5 and x = -2.
  • Completing the Square: We completed the square and arrived at the same roots, x = 5 and x = -2.

So, regardless of the method we choose, we consistently find that the roots of the equation y = x² - 3x - 10 are 5 and -2. This confirms that option B, "-2 and 5," is the correct answer.

Why Understanding Roots is Crucial

Finding the roots of a quadratic equation is not just an abstract mathematical exercise. It has practical applications in various fields. For instance, in physics, the roots of a quadratic equation can represent the time at which a projectile hits the ground. In engineering, they can represent the points of equilibrium in a system. In economics, they can represent the break-even points for a business. Understanding the concept of roots and mastering the methods for finding them is therefore a crucial skill for anyone pursuing a career in science, technology, engineering, or mathematics (STEM).

Common Mistakes to Avoid

When solving quadratic equations, it's easy to make mistakes if you're not careful. Here are some common pitfalls to watch out for:

  • Sign Errors: Pay close attention to the signs of the coefficients a, b, and c, especially when using the quadratic formula. A simple sign error can lead to incorrect roots.
  • Incorrect Factoring: Make sure you've factored the quadratic expression correctly. Double-check that the factors multiply back to the original expression.
  • Misapplying the Quadratic Formula: Ensure you're plugging the values of a, b, and c into the correct places in the formula. It's helpful to write out the formula and then substitute the values step-by-step.
  • Forgetting the ± Sign: Remember that the quadratic formula gives you two possible solutions due to the ± sign. Don't forget to consider both possibilities.
  • Arithmetic Errors: Be careful with your arithmetic calculations, especially when dealing with square roots and fractions. It's a good idea to double-check your work to avoid mistakes.

Tips for Mastering Quadratic Equations

Mastering quadratic equations takes practice and a solid understanding of the underlying concepts. Here are some tips to help you on your journey:

  • Practice Regularly: The more you practice solving quadratic equations, the more comfortable you'll become with the different methods and techniques.
  • Understand the Concepts: Don't just memorize formulas and procedures. Strive to understand why they work and how they're derived.
  • Choose the Right Method: Learn to recognize when factoring is the most efficient method and when the quadratic formula or completing the square is more appropriate.
  • Check Your Answers: Always check your solutions by plugging them back into the original equation to make sure they satisfy it.
  • Seek Help When Needed: Don't hesitate to ask for help from your teacher, classmates, or online resources if you're struggling with a particular concept or problem.

Conclusion: You've Got This!

So, there you have it! We've explored the fascinating world of quadratic equations, delved into the concept of roots, and mastered the methods for finding them. We've seen how factoring, the quadratic formula, and completing the square can all be used to solve the equation y = x² - 3x - 10, and we've confirmed that the roots are indeed -2 and 5. Remember, practice makes perfect, so keep working at it, and you'll be solving quadratic equations like a pro in no time. And remember, guys, math can be fun, especially when you conquer a challenging topic like this! Keep exploring, keep learning, and keep those mathematical gears turning! You've got this!