Solving Quadratic Equations: Find X In 2x^2 + 3x - 7

Are you wrestling with the equation 2x^2 + 3x - 7 = x^2 + 5x + 39 and trying to figure out the value of x? You're not alone! Quadratic equations can seem daunting, but with a systematic approach, they become much easier to handle. This guide will walk you through each step to solve this equation, ensuring you understand the process along the way. So, grab your pencil and paper, and let’s dive into the world of quadratic equations!

Understanding Quadratic Equations

First off, what exactly is a quadratic equation? A quadratic equation is a polynomial equation of the second degree. This means it includes at least one term that is squared (in our case, x^2). The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. Recognizing this form is the first step in solving any quadratic equation. In our given equation, we’ll need to rearrange it to fit this standard form before we can apply our solving methods.

Why are quadratic equations so important? They pop up in various real-world applications, from physics (think projectile motion) to engineering (designing structures) and even economics (modeling growth). Mastering the art of solving these equations not only boosts your math skills but also equips you with a powerful tool for problem-solving in numerous fields. We often use techniques such as factoring, completing the square, or the quadratic formula to find the solutions, which are also known as roots or zeros of the equation. Each method has its advantages, and the best approach often depends on the specific equation you're tackling.

To further illustrate, consider a simple example like x^2 - 4 = 0. This can be easily solved by adding 4 to both sides and then taking the square root, yielding x = ±2. However, more complex equations, like the one we're tackling today, require a bit more finesse. This is where understanding the underlying principles and the different solution methods becomes crucial. Our goal is not just to find the answer but to understand the process, so you'll be well-prepared to tackle any quadratic equation that comes your way.

Step 1: Rearrange the Equation

The initial equation we need to tackle is 2x^2 + 3x - 7 = x^2 + 5x + 39. To effectively solve for x, the very first step is to bring all the terms to one side of the equation. This means we want to manipulate the equation so that it fits the standard form of a quadratic equation, which, as we discussed earlier, is ax^2 + bx + c = 0. This form allows us to easily identify the coefficients a, b, and c, which are crucial for applying various solution methods.

So, how do we do this? We start by subtracting x^2 from both sides of the equation. This gives us x^2 + 3x - 7 = 5x + 39. Next, we subtract 5x from both sides, which leads to x^2 - 2x - 7 = 39. Finally, we subtract 39 from both sides to set the equation to zero, resulting in our rearranged equation: x^2 - 2x - 46 = 0. This is a critical step because it sets the stage for applying the quadratic formula or completing the square, which are our primary tools for finding the values of x.

The importance of this step cannot be overstated. Without properly rearranging the equation, we would be working with a non-standard form, making it incredibly difficult to apply the standard solution techniques. This rearrangement is like laying the foundation for a house; it must be solid and correct for everything else to build upon it. By ensuring that our equation is in the form ax^2 + bx + c = 0, we’ve made our task much more manageable and have paved the way for a successful solution. With the equation now in its standard form, we can move forward with confidence to the next steps in solving for x.

Step 2: Applying the Quadratic Formula

Now that we have our quadratic equation in the standard form, x^2 - 2x - 46 = 0, we can move on to one of the most powerful tools in our arsenal: the quadratic formula. The quadratic formula is a universal solution for any quadratic equation and is given by: x = (-b ± √(b^2 - 4ac)) / (2a). This formula might look a bit intimidating at first glance, but it’s a straightforward plug-and-chug method once you understand the components.

Before we dive into plugging in the values, let’s identify a, b, and c from our equation. In x^2 - 2x - 46 = 0, a is the coefficient of x^2, which is 1; b is the coefficient of x, which is -2; and c is the constant term, which is -46. Now we have all the pieces we need to use the quadratic formula. Substituting these values into the formula, we get: x = (-(-2) ± √((-2)^2 - 4 * 1 * (-46))) / (2 * 1). This might look like a mouthful, but it's just a matter of careful substitution.

Next, we simplify the expression step by step. First, -(-2) becomes 2. Then, we tackle the square root part: (-2)^2 is 4, and 4 * 1 * (-46) is -184. So, inside the square root, we have 4 - (-184), which simplifies to 4 + 184 = 188. Our equation now looks like: x = (2 ± √188) / 2. The square root of 188 can be simplified further. Since 188 = 4 * 47, we can rewrite √188 as √(4 * 47) = 2√47. Thus, our equation becomes: x = (2 ± 2√47) / 2.

Finally, we can simplify the entire expression by dividing both terms in the numerator by 2, which gives us: x = 1 ± √47. This is our solution for x. The ± sign indicates that there are two possible values for x: one where we add the square root and one where we subtract it. This comprehensive application of the quadratic formula has led us to our solution, and understanding each step ensures that you can confidently tackle similar problems in the future.

Step 3: Simplify the Solution

After applying the quadratic formula, we arrived at the solution x = 1 ± √47. This form is already quite simplified, but it's crucial to understand what this solution means and how it fits into the context of the original problem. The ± symbol indicates that we actually have two distinct solutions for x: one where we add the square root of 47 to 1, and another where we subtract it. These two values are the roots or zeros of the quadratic equation, which are the points where the parabola represented by the equation intersects the x-axis.

To fully appreciate the solution, let's break it down into its two separate values. The first solution is x = 1 + √47, and the second solution is x = 1 - √47. While these values might seem abstract, they are precise numerical answers. If you were to approximate √47, which is roughly 6.856, you would find that the two solutions are approximately x ≈ 7.856 and x ≈ -5.856. These approximations can be helpful for visualizing the solutions on a number line or graphing the quadratic equation.

It’s also important to note that √47 cannot be simplified further because 47 is a prime number. This means it has no factors other than 1 and itself, so we cannot extract any perfect squares from it. Therefore, the solution x = 1 ± √47 is in its simplest form. Understanding this level of simplification is key to presenting your answer correctly and demonstrating a solid grasp of mathematical principles.

In many practical applications, you might need to use these solutions for further calculations or interpretations. For example, if this equation represented a physical system, the solutions might correspond to specific points in time or space. Knowing how to extract and interpret these solutions is a crucial skill in mathematics and related fields. So, while we have simplified the solution to its final form, understanding its implications and how to use it in different contexts is the ultimate goal.

Conclusion

In conclusion, we successfully solved the quadratic equation 2x^2 + 3x - 7 = x^2 + 5x + 39 by methodically following a series of steps. We began by rearranging the equation into the standard form, which is crucial for applying the quadratic formula. This rearrangement involved bringing all terms to one side of the equation, resulting in x^2 - 2x - 46 = 0. Then, we identified the coefficients a, b, and c, which were essential for plugging into the quadratic formula.

Next, we applied the quadratic formula, x = (-b ± √(b^2 - 4ac)) / (2a), substituting the values a = 1, b = -2, and c = -46. This led us to x = (2 ± √188) / 2. We simplified the square root of 188 by recognizing that it could be expressed as 2√47, and then we further simplified the entire expression to obtain the final solution: x = 1 ± √47. This solution represents two distinct values for x, one obtained by adding √47 to 1 and the other by subtracting it.

Understanding each step of this process is vital for mastering quadratic equations. The ability to rearrange equations, apply the quadratic formula, and simplify solutions are fundamental skills in algebra and beyond. Moreover, knowing how to interpret these solutions in various contexts enhances your problem-solving capabilities in numerous fields. Quadratic equations are not just abstract mathematical concepts; they are powerful tools for modeling and solving real-world problems.

By following this step-by-step guide, you've not only found the solution to this specific equation but also gained a deeper understanding of how to approach and solve quadratic equations in general. This understanding will serve you well in future mathematical endeavors. If you want to learn more about quadratic equations and their applications, consider exploring resources like Khan Academy's Quadratic Equations Section.