Solving The Quadratic Equation 2n^2 - 7n + 6 = 0 A Step-by-Step Guide

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Hey guys! Today, we're diving into the fascinating world of quadratic equations, specifically tackling the equation 2n² - 7n + 6 = 0. Our mission? To find the possible values of n that make this equation true. We'll break it down step by step, so you can follow along easily. Quadratic equations are a cornerstone of algebra, popping up in various fields like physics, engineering, and even economics. Mastering them is crucial for anyone looking to excel in mathematics and related disciplines. So, let's get started and unlock the secrets hidden within this equation!

Understanding Quadratic Equations

Before we jump into solving our specific equation, let's take a moment to understand what quadratic equations are all about. A quadratic equation is essentially a polynomial equation of the second degree. What does that mean? Well, it means the highest power of the variable (in our case, n) is 2. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants (numbers), and a is not equal to zero (otherwise, it wouldn't be quadratic anymore!). These constants play a crucial role in determining the shape and position of the parabola, which is the graph of a quadratic function. The solutions to a quadratic equation, also known as roots or zeros, represent the points where the parabola intersects the x-axis. These roots are the values of the variable that make the equation equal to zero. Finding these roots is a fundamental skill in algebra and is used in various applications, from optimizing designs in engineering to modeling population growth in biology. Understanding the structure and properties of quadratic equations is the first step towards mastering their solutions. Now that we've got a handle on the basics, let's move on to the different methods we can use to solve them.

Methods for Solving Quadratic Equations

Alright, so how do we actually solve these quadratic equations? There are several methods in our arsenal, each with its own strengths and when it's best applied. We'll focus on two main methods that are particularly useful: factoring and the quadratic formula. Factoring is like reverse-engineering multiplication. We try to break down the quadratic expression into two simpler expressions (factors) that, when multiplied together, give us the original expression. This method is super efficient when it works, but it's not always straightforward. The quadratic formula, on the other hand, is a more general tool. It's a formula that provides the solutions directly, regardless of whether the equation can be factored easily or not. The quadratic formula is derived from the process of completing the square and guarantees a solution for any quadratic equation. In addition to these methods, completing the square is another powerful technique that transforms the quadratic equation into a perfect square trinomial, making it easier to solve. Each method offers a unique approach to finding the roots of a quadratic equation, and understanding when to use each one can significantly improve problem-solving efficiency. Let's dive deeper into factoring since it's the method we'll be using for our specific equation.

Solving by Factoring: A Detailed Look

Let's talk about factoring. This method is all about breaking down the quadratic expression into two binomials. Think of it like this: we're trying to find two expressions that, when multiplied, give us back our original quadratic equation. For our equation, 2n² - 7n + 6 = 0, the goal is to rewrite the left side as (something)(something else)* = 0. This allows us to use the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. Factoring involves identifying the coefficients of the quadratic equation and finding two numbers that satisfy specific conditions related to these coefficients. The process typically involves finding two numbers that multiply to give the product of the leading coefficient (the coefficient of the n² term) and the constant term, and that add up to the coefficient of the n term. Once these numbers are found, the middle term of the quadratic equation can be rewritten using these numbers, allowing the expression to be factored by grouping. Factoring is not only a method for solving quadratic equations but also a fundamental skill in simplifying algebraic expressions and solving various mathematical problems. In the next section, we'll apply this method step-by-step to solve our equation.

Step-by-Step Solution for 2n² - 7n + 6 = 0

Okay, guys, let's get our hands dirty and solve 2n² - 7n + 6 = 0 by factoring! Here's the breakdown:

  1. Identify the coefficients: In our equation, a = 2, b = -7, and c = 6. These coefficients are the key to unlocking the factors of the quadratic equation. Understanding the relationship between these coefficients and the factors is crucial for mastering the factoring technique. The coefficient a determines the stretch or compression of the parabola, b influences the position of the axis of symmetry, and c represents the y-intercept of the graph. These values not only help in solving the equation but also provide insights into the behavior of the quadratic function. Recognizing these connections can enhance your problem-solving abilities and deepen your understanding of quadratic equations.
  2. Find two numbers that multiply to ac and add up to b: We need two numbers that multiply to (2)(6) = 12 and add up to -7. After a bit of thought (and maybe some trial and error!), we find that -3 and -4 fit the bill perfectly. These numbers are the key to rewriting the middle term of the quadratic equation in a way that allows for factoring by grouping. The ability to quickly identify these numbers is a valuable skill in factoring, and it often comes with practice and familiarity with number properties. Mastering this step can significantly reduce the time and effort required to solve quadratic equations by factoring.
  3. Rewrite the middle term: Now, we rewrite the -7n term as -3n - 4n. Our equation becomes 2n² - 3n - 4n + 6 = 0. This step is the core of the factoring by grouping technique. By splitting the middle term into two terms with coefficients that satisfy the conditions identified in the previous step, we set up the expression for further simplification. This technique allows us to group terms in a way that common factors can be extracted, leading to the factorization of the quadratic expression. The choice of rewriting the middle term is not arbitrary; it is a strategic move that leverages the properties of the coefficients to facilitate factoring.
  4. Factor by grouping: Group the terms and factor out the greatest common factor (GCF) from each group: n(2n - 3) - 2(2n - 3) = 0. This step involves identifying common factors within the grouped terms and extracting them. Factoring out the GCF not only simplifies the expression but also reveals a common binomial factor, which is essential for the next step in the factoring process. The ability to factor by grouping is a valuable skill in algebra, and it is applicable not only to quadratic equations but also to other types of polynomial expressions.
  5. Factor out the common binomial: Notice that both terms have (2n - 3) in common. Factor it out: (2n - 3)(n - 2) = 0. This step is the culmination of the factoring process. By factoring out the common binomial factor, we have successfully expressed the quadratic equation as a product of two binomials. This form of the equation is crucial for applying the zero-product property, which allows us to find the solutions to the equation. Recognizing and extracting the common binomial factor is a critical skill in factoring and requires a keen eye for patterns and algebraic relationships.
  6. Apply the zero-product property: If (2n - 3)(n - 2) = 0, then either 2n - 3 = 0 or n - 2 = 0. The zero-product property is a fundamental principle in algebra that states that if the product of two factors is zero, then at least one of the factors must be zero. This property allows us to transform the problem of solving a single quadratic equation into the problem of solving two simpler linear equations. Understanding and applying the zero-product property is essential for solving equations by factoring, and it is a key concept in various areas of mathematics.
  7. Solve for n:
    • If 2n - 3 = 0, then 2n = 3, and n = 3/2. This step involves solving a linear equation for n. By isolating n on one side of the equation, we find one of the solutions to the quadratic equation. The ability to solve linear equations is a fundamental skill in algebra, and it is essential for solving quadratic equations by factoring. The solution n = 3/2 represents one of the roots of the quadratic equation, which corresponds to a point where the parabola intersects the x-axis.
    • If n - 2 = 0, then n = 2. Similarly, solving the second linear equation, we find another solution for n. This solution represents another root of the quadratic equation, and it corresponds to another point where the parabola intersects the x-axis. Finding both roots of the quadratic equation provides a complete solution to the problem and allows for a deeper understanding of the behavior of the quadratic function.

So, the possible values of n are 3/2 and 2!

The Answer and Why It's Correct

Therefore, the correct answer is a) n = 3/2, n = 2. We arrived at this answer by meticulously factoring the quadratic equation and applying the zero-product property. Factoring allowed us to break down the complex equation into simpler components, making it easier to isolate the variable n and find its possible values. Each step in the process, from identifying the coefficients to applying the zero-product property, was crucial in reaching the correct solution. Understanding the underlying principles of factoring and quadratic equations is essential for confidently solving these types of problems. The solutions we found, n = 3/2 and n = 2, are the values that make the quadratic equation equal to zero, and they represent the points where the parabola intersects the x-axis. This visual representation adds another layer of understanding to the solution, connecting the algebraic and graphical aspects of quadratic equations.

Common Mistakes to Avoid

Let's chat about some common pitfalls when solving quadratic equations, so you can dodge them! One frequent mistake is messing up the signs when factoring. Double-check those signs, guys! Also, sometimes people forget to set each factor to zero after factoring. Remember that zero-product property! Another common error is attempting to factor when it's not the most efficient method. While factoring is great when it works, it can be time-consuming or even impossible for some quadratic equations. In such cases, using the quadratic formula or completing the square might be more appropriate. Understanding the strengths and limitations of each method is crucial for efficient problem-solving. Additionally, neglecting to simplify the solutions or providing only one solution when there are two is a common mistake. Always remember that a quadratic equation can have up to two distinct solutions. By being mindful of these common errors, you can significantly improve your accuracy and confidence in solving quadratic equations.

Conclusion: Mastering Quadratic Equations

Woohoo! We've successfully navigated the quadratic equation 2n² - 7n + 6 = 0 and found the possible values of n. Remember, guys, practice makes perfect! The more you work with quadratic equations, the more comfortable and confident you'll become. Quadratic equations are a fundamental part of mathematics and have wide-ranging applications in various fields. Mastering the techniques for solving them, such as factoring and using the quadratic formula, is essential for success in algebra and beyond. Don't be afraid to tackle challenging problems and explore different methods. The journey of learning mathematics is a rewarding one, and each solved equation brings you closer to a deeper understanding of the subject. So, keep practicing, keep exploring, and keep enjoying the beauty of mathematics!