Solving 3x² - 12x + 9 = 0 A Step-by-Step Quadratic Equation Guide
Introduction to Quadratic Equations
In the realm of mathematics, quadratic equations hold a position of paramount importance. These equations, characterized by their highest power of the variable being two, frequently appear in diverse fields such as physics, engineering, economics, and computer science. Mastering the techniques to solve quadratic equations is not just an academic exercise; it's a fundamental skill that unlocks the door to understanding and modeling real-world phenomena. This guide delves into a systematic approach for solving a specific quadratic equation: 3x² - 12x + 9 = 0. By breaking down each step and explaining the underlying principles, we aim to equip you with the knowledge and confidence to tackle similar problems.
A quadratic equation is generally expressed in the form ax² + bx + c = 0, where a, b, and c are constants, and 'a' is not equal to zero. The solutions to a quadratic equation are the values of 'x' that satisfy the equation, often referred to as the roots or zeros of the equation. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. Each method has its strengths and weaknesses, and the choice of method often depends on the specific form of the equation. For instance, factoring is a preferred method when the equation can be easily factored into two linear expressions. Completing the square is a more versatile method that can be applied to any quadratic equation, although it may involve more steps. The quadratic formula provides a direct solution and is particularly useful when factoring is difficult or impossible.
Before diving into the solution of our specific equation, it's crucial to understand the underlying concepts. The graph of a quadratic equation is a parabola, and the roots of the equation correspond to the x-intercepts of the parabola. The discriminant, which is the part of the quadratic formula under the square root (b² - 4ac), provides valuable information about the nature of the roots. If the discriminant is positive, the equation has two distinct real roots; if it's zero, the equation has one real root (a repeated root); and if it's negative, the equation has two complex roots. Understanding these concepts not only helps in solving quadratic equations but also provides a deeper insight into their behavior and applications. In the following sections, we will apply these concepts to solve the equation 3x² - 12x + 9 = 0, demonstrating each step in detail and explaining the reasoning behind it. By the end of this guide, you will have a solid understanding of how to solve quadratic equations and be able to apply these skills to a variety of problems.
Simplifying the Equation
When faced with the quadratic equation 3x² - 12x + 9 = 0, our first step is to simplify the equation. Simplification makes the equation easier to handle and reduces the complexity of the subsequent steps. In this case, we observe that all the coefficients (3, -12, and 9) are divisible by 3. This suggests that we can divide the entire equation by 3 to obtain a simpler equivalent equation. Dividing both sides of the equation by 3, we get:
(3x² - 12x + 9) / 3 = 0 / 3
This simplifies to:
x² - 4x + 3 = 0
This simplified quadratic equation is much easier to work with than the original equation. The coefficients are smaller, and the factoring process, if applicable, becomes more straightforward. This step of simplification is a crucial aspect of problem-solving in mathematics. By identifying common factors and dividing them out, we can reduce the computational burden and make the problem more manageable. In general, whenever you encounter a quadratic equation where the coefficients have a common factor, it's advisable to simplify the equation first. This not only makes the subsequent steps easier but also reduces the chances of making errors in calculations.
Simplifying the equation also helps in visualizing the problem more clearly. The simplified form often reveals patterns and relationships that might not be apparent in the original equation. For instance, in this case, the simplified equation x² - 4x + 3 = 0 makes it easier to identify potential factors that might lead to a solution. This ability to simplify and visualize problems is a key skill in mathematical problem-solving. It allows us to break down complex problems into smaller, more manageable parts and to approach them in a systematic manner. In the next section, we will explore different methods for solving this simplified quadratic equation, including factoring, completing the square, and using the quadratic formula. Each method has its advantages and disadvantages, and the choice of method often depends on the specific characteristics of the equation. However, the initial step of simplification remains a crucial part of the process, regardless of the method chosen.
Method 1: Factoring the Quadratic Equation
One of the most efficient methods for solving quadratic equations is factoring. Factoring involves expressing the quadratic equation as a product of two linear factors. This method is particularly effective when the quadratic equation can be easily factored. In our case, the simplified equation is x² - 4x + 3 = 0. To factor this equation, we need to find two numbers that multiply to 3 (the constant term) and add up to -4 (the coefficient of the x term). By considering the factors of 3, which are 1 and 3, and their negative counterparts, -1 and -3, we can see that -1 and -3 satisfy both conditions:
(-1) * (-3) = 3
(-1) + (-3) = -4
Therefore, we can factor the quadratic equation as follows:
x² - 4x + 3 = (x - 1)(x - 3) = 0
Now, according to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. This gives us two separate linear equations:
x - 1 = 0 or x - 3 = 0
Solving these equations for x, we get:
x = 1 or x = 3
These are the solutions to the quadratic equation x² - 4x + 3 = 0. Factoring is a powerful technique for solving quadratic equations because it directly leads to the solutions once the factors are identified. However, not all quadratic equations can be easily factored. In such cases, other methods like completing the square or the quadratic formula may be more appropriate.
The success of factoring depends on our ability to quickly identify the factors that satisfy the required conditions. Practice and familiarity with factoring techniques are essential for mastering this method. When factoring, it's helpful to consider the signs of the coefficients and the constant term. For example, in our equation, the positive constant term and the negative coefficient of the x term suggest that both factors will involve negative numbers. This kind of reasoning can help narrow down the possibilities and make the factoring process more efficient. In the next section, we will explore another method for solving quadratic equations, completing the square, which is a more general method that can be applied to any quadratic equation, regardless of whether it can be easily factored.
Method 2: Using the Quadratic Formula
When factoring proves difficult or impossible, the quadratic formula provides a reliable alternative for solving quadratic equations. The quadratic formula is a general formula that gives the solutions to any quadratic equation in the form ax² + bx + c = 0. The formula is:
x = (-b ± √(b² - 4ac)) / (2a)
To apply the quadratic formula to our simplified equation x² - 4x + 3 = 0, we first identify the coefficients: a = 1, b = -4, and c = 3. Substituting these values into the quadratic formula, we get:
x = (-(-4) ± √((-4)² - 4 * 1 * 3)) / (2 * 1)
Simplifying the expression under the square root:
x = (4 ± √(16 - 12)) / 2
x = (4 ± √4) / 2
x = (4 ± 2) / 2
This gives us two possible solutions:
x₁ = (4 + 2) / 2 = 6 / 2 = 3
x₂ = (4 - 2) / 2 = 2 / 2 = 1
Thus, the solutions to the quadratic equation x² - 4x + 3 = 0, obtained using the quadratic formula, are x = 1 and x = 3. These are the same solutions we found by factoring, which confirms the correctness of both methods. The quadratic formula is a powerful tool because it can be applied to any quadratic equation, regardless of its factorability. It is particularly useful when the coefficients are large or when the equation has irrational or complex roots.
One of the key components of the quadratic formula is the discriminant, b² - 4ac, which appears under the square root. The discriminant provides valuable information about the nature of the roots. As mentioned earlier, if the discriminant is positive, the equation has two distinct real roots; if it's zero, the equation has one real root (a repeated root); and if it's negative, the equation has two complex roots. In our case, the discriminant is 4, which is positive, indicating that the equation has two distinct real roots, as we found. The quadratic formula is a cornerstone of algebra and is essential for anyone studying mathematics, physics, engineering, or any field that involves mathematical modeling. In the next section, we will summarize our findings and discuss the importance of understanding different methods for solving quadratic equations.
Summary and Conclusion
In this comprehensive guide, we have explored the process of solving the quadratic equation 3x² - 12x + 9 = 0. We began by simplifying the equation by dividing all terms by 3, resulting in the simpler equivalent equation x² - 4x + 3 = 0. This simplification step is crucial as it makes the equation easier to handle and reduces the complexity of the subsequent steps. We then employed two distinct methods to solve the equation: factoring and the quadratic formula.
Factoring involves expressing the quadratic equation as a product of two linear factors. In our case, we found that x² - 4x + 3 = (x - 1)(x - 3), which led to the solutions x = 1 and x = 3. Factoring is an efficient method when the quadratic equation can be easily factored, but it may not be applicable to all quadratic equations.
The quadratic formula, on the other hand, is a general formula that can be used to solve any quadratic equation in the form ax² + bx + c = 0. By substituting the coefficients of our simplified equation into the formula, we obtained the same solutions, x = 1 and x = 3. The quadratic formula is a powerful tool that guarantees a solution, even when factoring is difficult or impossible.
Understanding different methods for solving quadratic equations is essential for several reasons. First, it provides flexibility in problem-solving. Some methods may be more suitable for certain types of equations, and having a repertoire of techniques allows you to choose the most efficient approach. Second, it deepens your understanding of the underlying mathematical principles. Each method offers a different perspective on the nature of quadratic equations and their solutions. Finally, it builds confidence in your problem-solving abilities. By mastering different methods, you become more adept at tackling a wide range of mathematical problems.
In conclusion, solving quadratic equations is a fundamental skill in mathematics with applications in various fields. By understanding the different methods available and practicing their application, you can develop a strong foundation in algebra and enhance your problem-solving skills. The equation 3x² - 12x + 9 = 0 serves as a valuable example for illustrating these techniques, and the principles discussed in this guide can be applied to a wide range of quadratic equations and related problems. Remember to always simplify equations whenever possible, choose the most appropriate method for the given problem, and practice regularly to improve your skills and understanding.
