Solving $5x^2 - 30x + 45 = 0$: A Step-by-Step Guide

Let's dive into solving the quadratic equation $5x^2 - 30x + 45 = 0$. Quadratic equations might seem intimidating at first, but with a systematic approach, they can be tackled with ease. This article will guide you through each step, ensuring you understand the process thoroughly. So, grab a pen and paper, and let's get started!

1. Simplify the Equation

Before we jump into any formulas, it's always a good idea to simplify the equation if possible. Notice that all the coefficients in the equation $5x^2 - 30x + 45 = 0$ are divisible by 5. Dividing the entire equation by 5 will make our calculations much easier. This is a crucial initial step because it reduces the complexity of the numbers we're dealing with, making subsequent steps less prone to errors and easier to manage. By performing this simplification, we transform the equation into a more manageable form without altering its fundamental properties or solutions. Simplifying at this stage allows us to work with smaller, more convenient numbers, leading to a smoother and more efficient solution process. Remember, the goal is to make the equation as straightforward as possible before applying more complex methods, ensuring accuracy and saving time in the long run. Recognizing opportunities for simplification is a key skill in solving mathematical problems, and it often leads to a clearer path to the correct answer. In this case, dividing by 5 sets us up for success by streamlining the equation and preparing it for further analysis.

Dividing each term by 5, we get:

$x^2 - 6x + 9 = 0$

2. Factoring the Quadratic

Now that we have a simplified quadratic equation, $x^2 - 6x + 9 = 0$, the next step is to factor it. Factoring involves expressing the quadratic expression as a product of two binomials. To do this, we look for two numbers that multiply to give the constant term (9) and add up to give the coefficient of the linear term (-6). This process requires careful consideration of the signs and factors of the constant term. In this case, the numbers are -3 and -3 because (-3) * (-3) = 9 and (-3) + (-3) = -6. Factoring is an essential technique for solving quadratic equations because it transforms the equation into a form where we can easily identify the solutions. By breaking down the quadratic expression into its constituent factors, we can apply 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 property allows us to set each factor equal to zero and solve for the variable, providing us with the roots of the quadratic equation. Mastering the skill of factoring enables us to solve a wide range of quadratic equations efficiently and accurately. It's a fundamental concept in algebra that builds a solid foundation for more advanced mathematical topics.

We can rewrite the equation as:

$(x - 3)(x - 3) = 0$

Which can also be written as:

$(x - 3)^2 = 0$

3. Solve for x

With the equation in factored form, $(x - 3)^2 = 0$, we can now easily solve for x. To do this, we take the square root of both sides of the equation. This step is crucial because it isolates the variable x and allows us to determine its value. When we take the square root of a squared term, we essentially undo the squaring operation, leaving us with the original term. In this case, taking the square root of $(x - 3)^2$ gives us $(x - 3)$. Similarly, taking the square root of 0 gives us 0. This simplification leads to a straightforward equation that we can solve directly for x. This method works because the square root function is the inverse operation of squaring, allowing us to reverse the process and find the value of x that satisfies the original equation. By understanding and applying this technique, we can efficiently solve quadratic equations that are expressed in factored form, making it a valuable tool in our mathematical problem-solving toolkit. Recognizing and utilizing inverse operations is a fundamental concept in algebra that extends to solving various types of equations and inequalities.

Taking the square root of both sides:

$x - 3 = 0$

Adding 3 to both sides:

$x = 3$

4. Verify the Solution

To ensure our solution is correct, it's always a good practice to verify it by substituting x = 3 back into the original equation, $5x^2 - 30x + 45 = 0$. This step is essential because it confirms that the value we found for x satisfies the equation. By substituting x = 3 into the equation, we can evaluate both sides and check if they are equal. If the equation holds true, then our solution is correct. However, if the equation does not hold true, it indicates that there may be an error in our calculations, and we need to revisit our steps to identify and correct the mistake. Verifying the solution is a crucial step in problem-solving because it helps us avoid errors and ensures the accuracy of our results. It also reinforces our understanding of the equation and the solution process. This practice is particularly important in complex problems where there are multiple steps involved, as it allows us to catch any mistakes early on and prevent them from propagating through the rest of the solution. Always take the time to verify your solutions, as it's a valuable habit that will improve your accuracy and confidence in solving mathematical problems.

Substituting x = 3:

$5(3)^2 - 30(3) + 45 = 5(9) - 90 + 45 = 45 - 90 + 45 = 0$

Since the equation holds true, our solution is correct.

Conclusion

The solution to the quadratic equation $5x^2 - 30x + 45 = 0$ is x = 3. By simplifying the equation, factoring it, and solving for x, we were able to find the solution. Always remember to verify your solution to ensure accuracy. Understanding how to solve quadratic equations is a fundamental skill in algebra, and with practice, you'll become more proficient at it. Keep practicing and exploring different types of quadratic equations to enhance your problem-solving abilities. Solving quadratic equations is a cornerstone of algebra, and mastering this skill opens doors to more advanced mathematical concepts and applications. With a solid understanding of the techniques involved, you'll be well-equipped to tackle a wide range of problems in various fields, from physics and engineering to economics and computer science. Embrace the challenge and continue to hone your skills, and you'll find that quadratic equations become less daunting and more manageable over time. Remember, practice makes perfect, so keep exploring and experimenting with different types of equations to broaden your mathematical horizons and deepen your understanding of the subject.

For further learning, you can check out resources like Khan Academy for more examples and practice problems.