Solving Quadratic Equations Step-by-Step Guide
Hey guys! Today, we're diving into a common math problem: solving a quadratic equation. Specifically, we'll tackle the equation (7/2)x^2 = 126. Don't worry, it's not as scary as it looks! We'll break it down step by step, so even if math isn't your favorite subject, you'll be able to follow along. Our goal is to find the value(s) of 'x' that make this equation true. So, grab your thinking caps, and let's get started!
Understanding Quadratic Equations
Before we jump into solving our specific equation, let's quickly recap what quadratic equations are all about. Quadratic equations are polynomial equations of the second degree. This basically means the highest power of the variable (in our case, 'x') is 2. The general form of a quadratic equation is ax^2 + bx + c = 0, where 'a', 'b', and 'c' are constants (numbers). Our equation, (7/2)x^2 = 126, is a simplified version where 'b' is 0 and we have a constant term on the other side of the equation.
Why are quadratic equations important? Well, they pop up everywhere in the real world! From calculating the trajectory of a ball thrown in the air to designing bridges and buildings, quadratic equations play a crucial role in many fields of science and engineering. Mastering how to solve them is a fundamental skill in mathematics. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. However, for this particular equation, we'll use a more direct algebraic approach, which is the most efficient method when dealing with equations in this format. This method involves isolating the x^2 term and then taking the square root of both sides.
Understanding the structure of quadratic equations helps in recognizing the most suitable method for solving them. For instance, if the equation can be easily factored, that might be the quickest route. If completing the square seems straightforward, that's another option. And, of course, the quadratic formula is a reliable method that works for any quadratic equation. Each method has its strengths and weaknesses, and choosing the right one can save you time and effort. Remember, practice makes perfect! The more you work with quadratic equations, the better you'll become at identifying the best solution method. We're focusing on a specific, simplified case here, which allows us to use a very direct approach. This not only gives us the answer quickly but also helps build a solid understanding of the underlying principles.
Step-by-Step Solution
Okay, let's get down to business and solve our equation: (7/2)x^2 = 126. Remember, our goal is to isolate 'x' on one side of the equation. Here's how we'll do it:
Step 1: Isolate the x^2 term
Our first step is to get the x^2 term by itself. To do this, we need to get rid of the fraction (7/2) that's multiplying it. The easiest way to do that is to multiply both sides of the equation by the reciprocal of (7/2), which is (2/7). This will cancel out the fraction on the left side. So, we have:
(2/7) * (7/2)x^2 = 126 * (2/7)
The (2/7) and (7/2) on the left side cancel out, leaving us with:
x^2 = 126 * (2/7)
Now, let's simplify the right side. We can multiply 126 by 2 to get 252, and then divide by 7:
x^2 = 252 / 7
x^2 = 36
Step 2: Take the square root of both sides
Now we have x^2 = 36. To find 'x', we need to take the square root of both sides of the equation. Remember, when we take the square root, we need to consider both the positive and negative solutions. This is because both a positive and a negative number, when squared, will give us a positive result. So:
√x^2 = ±√36
This simplifies to:
x = ±6
So, we have two solutions: x = 6 and x = -6.
Step 3: Verify the solutions
It's always a good idea to check our answers to make sure they work. Let's plug both solutions back into the original equation and see if they hold true:
For x = 6:
(7/2) * (6)^2 = 126
(7/2) * 36 = 126
7 * 18 = 126
126 = 126 (This is true!)
For x = -6:
(7/2) * (-6)^2 = 126
(7/2) * 36 = 126
7 * 18 = 126
126 = 126 (This is also true!)
Both solutions work, so we've successfully solved the equation!
Why Two Solutions?
You might be wondering why we have two solutions for this quadratic equation. The reason lies in the nature of squaring a number. As we saw, both 6 and -6, when squared, result in 36. This is a fundamental property of quadratic equations – they can have up to two real solutions. These solutions represent the points where the graph of the quadratic equation intersects the x-axis. In our case, the graph of y = (7/2)x^2 - 126 would intersect the x-axis at x = 6 and x = -6. Understanding this concept is crucial for grasping the broader implications of quadratic equations. They don't just give us numbers; they tell a story about the relationship between variables. The two solutions represent two specific points where a certain condition is met, in this case, where the value of the quadratic expression equals zero. This graphical and conceptual understanding enhances our ability to apply quadratic equations to real-world problems.
Alternative Methods (Brief Overview)
While we used a direct algebraic method for this particular equation, it's worth knowing that there are other ways to solve quadratic equations. As mentioned earlier, these include factoring, completing the square, and the quadratic formula. Factoring involves breaking down the quadratic expression into two binomials. This method is efficient when the equation can be easily factored, but it's not always applicable. Completing the square is a technique that transforms the quadratic equation into a perfect square trinomial, which can then be easily solved. This method is more versatile than factoring but can be a bit more involved. The quadratic formula is a universal method that works for any quadratic equation, regardless of its complexity. It's a powerful tool to have in your arsenal, but it can sometimes be more time-consuming than other methods. The choice of method often depends on the specific equation and your personal preference. Each method offers a unique approach to solving quadratic equations, and understanding them all provides a comprehensive toolkit for tackling these types of problems.
Real-World Applications
As we touched on earlier, quadratic equations aren't just abstract mathematical concepts; they have numerous applications in the real world. For example, they are used extensively in physics to model projectile motion. When you throw a ball, its path through the air can be described by a quadratic equation. The equation helps us determine how high the ball will go and how far it will travel. In engineering, quadratic equations are used in the design of bridges and arches. The curves in these structures often follow a parabolic shape, which is described by a quadratic equation. By understanding these equations, engineers can ensure the stability and safety of these structures. Quadratic equations also play a role in economics and finance. They can be used to model cost and revenue functions, helping businesses make informed decisions about pricing and production. From the simple act of throwing a ball to complex engineering projects and economic forecasting, quadratic equations are a fundamental tool for understanding and shaping the world around us. Their widespread use highlights the importance of mastering these mathematical concepts.
Practice Problems
Now that we've walked through a step-by-step solution and discussed the broader context of quadratic equations, it's time to put your knowledge to the test! Here are a few practice problems to help you solidify your understanding. Remember, the key to mastering any mathematical concept is practice, practice, practice. Work through these problems carefully, and don't be afraid to refer back to the steps we discussed earlier. If you get stuck, try breaking the problem down into smaller steps or reviewing the fundamental principles. Each problem is an opportunity to strengthen your skills and build confidence in your ability to solve quadratic equations.
- Solve for x: 3x^2 = 75
- Find the solutions for: (1/2)x^2 = 50
- What are the roots of: 2x^2 = 98
Work through these problems at your own pace, and remember to check your answers. Solving these practice problems will not only reinforce your understanding of the methods we've discussed but also help you develop your problem-solving skills. Each equation presents a unique opportunity to apply the concepts you've learned and refine your approach. Don't hesitate to explore different strategies and learn from any mistakes you make along the way. The process of working through these problems is just as important as the final answer. It's through this active engagement with the material that you'll truly master the art of solving quadratic equations.
Conclusion
So, there you have it! We've successfully solved the equation (7/2)x^2 = 126 and found that x = 6 and x = -6. We also took a broader look at quadratic equations, their importance, and their real-world applications. Hopefully, this step-by-step guide has made solving quadratic equations a little less intimidating and a lot more understandable. Remember, math is like any other skill – the more you practice, the better you'll get. Keep working at it, and you'll be a quadratic equation-solving pro in no time! Keep practicing, keep exploring, and most importantly, keep enjoying the journey of learning mathematics. Each equation you solve is a step forward in your mathematical journey. And remember, the skills you develop in math, such as problem-solving and logical thinking, are valuable assets in all aspects of life. So, embrace the challenge, celebrate your successes, and never stop learning!
