Solving 9x² = 4: A Step-by-Step Guide To Quadratic Equation Solutions
Introduction: Understanding Quadratic Equations
When dealing with equations in mathematics, quadratic equations hold a significant place due to their wide range of applications in various fields, from physics to engineering. A quadratic equation is a polynomial equation of the second degree, meaning it involves at least one term that is squared. 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 zero. Understanding how to solve these equations is crucial for anyone delving into algebra and beyond. This article aims to provide a comprehensive guide on solving a specific quadratic equation, 9x² = 4, while also enhancing your understanding of the underlying principles and methods involved.
Before diving into the specifics of our equation, it's essential to grasp the fundamental concepts of solving quadratic equations. The solutions to a quadratic equation are also known as its roots or zeros. These roots represent the values of x that satisfy the equation, making the left-hand side equal to the right-hand side. There are several methods to find these roots, each with its own advantages and applicability. The most common methods include factoring, using the quadratic formula, and completing the square. In this article, we will primarily focus on methods that are best suited for the given equation, 9x² = 4, which is a special case where the b term is zero. This allows for a more straightforward solution process, often involving isolating the squared term and then taking the square root.
Problem Statement: The Quadratic Equation 9x² = 4
The specific quadratic equation we aim to solve is 9x² = 4. This equation falls under the category of quadratic equations because it includes a term with x raised to the power of 2. However, it is a simplified form compared to the general quadratic equation ax² + bx + c = 0 because it lacks the bx term. This simplification makes it easier to solve directly without resorting to more complex methods like the quadratic formula or completing the square. Our goal is to find the values of x that, when substituted into the equation, make the equation true. In other words, we are looking for the roots of the equation. The absence of the bx term indicates that the roots will likely be symmetric around zero, which means if one solution is a positive number, the other solution will be its negative counterpart. This is a key observation that can guide our solution process.
Understanding the structure of the equation 9x² = 4 is crucial for choosing the most efficient solution method. The equation presents a direct relationship between the square of x and a constant, which suggests that isolating x² and then taking the square root is a viable approach. This method is particularly effective when the equation is in this simplified form, as it avoids the complexities of factoring or using the quadratic formula. By recognizing the specific characteristics of this equation, we can tailor our solution strategy to be both accurate and efficient. The following sections will delve into the step-by-step process of solving this equation, highlighting the mathematical principles involved and providing clear explanations for each step.
Method 1: Solving by Isolating x² and Taking the Square Root
One of the most straightforward methods to solve the quadratic equation 9x² = 4 is by isolating the term x² and then taking the square root of both sides. This approach is particularly effective when the equation is in a simplified form, lacking the linear term (bx). The first step in this method involves isolating x² on one side of the equation. To do this, we need to eliminate the coefficient 9 that is multiplying x². We can achieve this by dividing both sides of the equation by 9. This maintains the balance of the equation and moves us closer to isolating x².
Dividing both sides of the equation 9x² = 4 by 9, we get:
(9x²)/9 = 4/9
This simplifies to:
x² = 4/9
Now that we have isolated x², the next step is to take the square root of both sides of the equation. It's crucial to remember that when taking the square root, we must consider both the positive and negative roots. This is because both the positive and negative square roots, when squared, will result in the same positive value. In this case, we are looking for the numbers that, when squared, give us 4/9. Taking the square root of both sides, we get:
√(x²) = ±√(4/9)
This simplifies to:
x = ±(2/3)
Thus, we find two solutions for x: x = 2/3 and x = -2/3. This method demonstrates a direct and efficient way to solve quadratic equations when they are in a simplified form, highlighting the importance of recognizing the structure of the equation to choose the most appropriate solution strategy.
Method 2: Rewriting as a Difference of Squares and Factoring
Another effective method for solving the quadratic equation 9x² = 4 involves rewriting the equation and factoring it as a difference of squares. This technique leverages the algebraic identity a² - b² = (a + b)(a - b), which is a fundamental concept in algebra. To apply this method, we first need to rearrange the equation so that it is set equal to zero. This is a standard practice in solving quadratic equations, as it allows us to use factoring techniques more effectively.
Starting with the equation 9x² = 4, we subtract 4 from both sides to set the equation equal to zero:
9x² - 4 = 0
Now, we recognize that both 9x² and 4 are perfect squares. 9x² can be expressed as (3x)², and 4 can be expressed as 2². This allows us to rewrite the equation as a difference of squares:
(3x)² - 2² = 0
Using the difference of squares identity, a² - b² = (a + b)(a - b), we can factor the equation as follows:
(3x + 2)(3x - 2) = 0
Now that we have factored the equation, we can 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. This means that either (3x + 2) = 0 or (3x - 2) = 0. We can solve each of these linear equations separately.
For the first factor, 3x + 2 = 0, we subtract 2 from both sides:
3x = -2
Then, we divide by 3:
x = -2/3
For the second factor, 3x - 2 = 0, we add 2 to both sides:
3x = 2
Then, we divide by 3:
x = 2/3
Thus, we obtain the same solutions as in the previous method: x = 2/3 and x = -2/3. This method demonstrates the power of algebraic identities in simplifying and solving quadratic equations, providing a valuable alternative approach to direct isolation and square root extraction.
Comparing the Solutions and Choosing the Best Method
In the previous sections, we explored two distinct methods for solving the quadratic equation 9x² = 4: isolating x² and taking the square root, and rewriting the equation as a difference of squares and factoring. Both methods successfully yielded the same solutions: x = 2/3 and x = -2/3. However, the choice of method can impact the efficiency and clarity of the solution process. Understanding the strengths and weaknesses of each approach is crucial for developing problem-solving skills in mathematics.
The method of isolating x² and taking the square root is particularly efficient for equations in the form ax² = c, where there is no linear term (bx). This method directly addresses the structure of the equation, making it straightforward and quick. It involves fewer steps and is less prone to errors, especially for those who are comfortable with basic algebraic manipulations. The key advantage of this method is its simplicity and directness, making it an excellent choice for this specific type of quadratic equation.
On the other hand, the method of rewriting as a difference of squares and factoring is a more general technique that can be applied to a broader range of quadratic equations. It leverages the algebraic identity a² - b² = (a + b)(a - b), which is a fundamental concept in algebra. While this method is slightly more involved than the first method, it reinforces the importance of factoring and the zero-product property, which are essential skills for solving more complex quadratic equations. This method is particularly useful when the equation can be easily recognized as a difference of squares, providing a structured approach to finding the solutions.
In the case of 9x² = 4, the method of isolating x² and taking the square root is arguably the more efficient choice due to the equation's simple structure. However, understanding and practicing both methods is beneficial for developing a comprehensive understanding of quadratic equations and problem-solving strategies. Ultimately, the "best" method depends on the specific equation and the individual's familiarity and comfort with different algebraic techniques.
Verification of Solutions
After finding the solutions to a quadratic equation, it's essential to verify that these solutions are correct. This process involves substituting the solutions back into the original equation to ensure that they satisfy the equation. Verification not only confirms the accuracy of the solutions but also reinforces the understanding of what it means for a value to be a solution to an equation. For the quadratic equation 9x² = 4, we found two solutions: x = 2/3 and x = -2/3. We will now substitute each of these values back into the original equation to verify them.
First, let's verify the solution x = 2/3. Substituting this value into the equation 9x² = 4, we get:
9(2/3)² = 4
Squaring 2/3, we get:
9(4/9) = 4
Multiplying 9 by 4/9, we get:
4 = 4
This confirms that x = 2/3 is indeed a solution to the equation, as it makes the equation true.
Next, let's verify the solution x = -2/3. Substituting this value into the equation 9x² = 4, we get:
9(-2/3)² = 4
Squaring -2/3, we also get:
9(4/9) = 4
Multiplying 9 by 4/9, we get:
4 = 4
This also confirms that x = -2/3 is a solution to the equation. The fact that squaring a negative number results in a positive number is a key concept in understanding why both positive and negative roots can be solutions to quadratic equations.
By verifying both solutions, we have demonstrated that they accurately satisfy the original equation. This process underscores the importance of checking solutions in mathematical problem-solving, ensuring that the answers are correct and that the underlying concepts are well understood.
Conclusion: Mastering Quadratic Equation Solutions
In this article, we have thoroughly explored the process of solving the quadratic equation 9x² = 4. We examined two distinct methods: isolating x² and taking the square root, and rewriting the equation as a difference of squares and factoring. Both methods led us to the same solutions, x = 2/3 and x = -2/3, demonstrating the versatility of algebraic techniques in solving quadratic equations. We also emphasized the importance of verifying solutions to ensure accuracy and reinforce understanding.
Throughout our exploration, we highlighted the significance of recognizing the structure of a given equation to choose the most efficient solution method. For the equation 9x² = 4, the method of isolating x² and taking the square root proved to be the most direct and straightforward approach. However, understanding the method of factoring as a difference of squares provides a valuable alternative and reinforces fundamental algebraic principles. Mastering these methods equips individuals with a robust toolkit for tackling a wide range of quadratic equations.
Solving quadratic equations is a fundamental skill in algebra, with applications extending to various fields of mathematics and beyond. The ability to confidently and accurately solve these equations is crucial for success in higher-level mathematics courses and in practical applications in science, engineering, and other disciplines. By understanding the underlying principles and practicing different solution methods, one can develop a strong foundation in algebra and enhance their problem-solving abilities. The quadratic equation 9x² = 4 serves as an excellent example to illustrate these principles, offering a clear and concise case study for mastering quadratic equation solutions.
