Solving Quadratic Equations A Detailed Guide To 5x^2 - 7x - 18 = 0

by ADMIN 67 views

Hey everyone! Today, we're going to dive deep into solving a quadratic equation. Quadratic equations might seem intimidating at first, but trust me, with a step-by-step approach, you'll be able to tackle them like a pro. We'll be focusing on the equation 5x2−7x−18=05x^2 - 7x - 18 = 0, and by the end of this guide, you'll not only know the correct solution but also understand the underlying concepts and methods. So, let's get started!

Understanding Quadratic Equations

First, let's understand what we're dealing with. A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants, and xx is the variable we want to solve for. In our case, the equation is 5x2−7x−18=05x^2 - 7x - 18 = 0. Here, a=5a = 5, b=−7b = -7, and c=−18c = -18. Recognizing these coefficients is the first step in solving the equation. Quadratic equations can have two solutions, one solution, or no real solutions, depending on the discriminant, which we'll discuss later.

To find the solutions, also known as roots or zeros, we have several methods at our disposal. The most common methods are factoring, completing the square, and using the quadratic formula. For this particular equation, we'll focus on using the quadratic formula, as it's a reliable method for any quadratic equation. Before we jump into the formula, it's worth noting why understanding these methods is so important. Quadratic equations pop up in various real-world applications, from physics to engineering to finance. For example, they can be used to model projectile motion, calculate areas, and even predict financial growth. Mastering quadratic equations opens doors to solving a wide range of problems.

Now, let's talk about the quadratic formula itself. It's a powerful tool that gives us the solutions directly from the coefficients of the equation. The formula is: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where aa, bb, and cc are the coefficients we identified earlier. The ±\pm symbol means we have two possible solutions: one with addition and one with subtraction. The expression inside the square root, b2−4acb^2 - 4ac, is called the discriminant. The discriminant tells us about the nature of the solutions. If it's positive, we have two distinct real solutions; if it's zero, we have one real solution (a repeated root); and if it's negative, we have two complex solutions. This is a crucial concept to grasp, as it guides our expectations about the type of solutions we'll find. So, with our equation 5x2−7x−18=05x^2 - 7x - 18 = 0 and the quadratic formula in hand, we're well-equipped to find the solutions. Let's move on to applying the formula step by step.

Applying the Quadratic Formula to 5x2−7x−18=05x^2 - 7x - 18 = 0

Okay, let's put the quadratic formula into action! We have our equation, 5x2−7x−18=05x^2 - 7x - 18 = 0, and we've identified the coefficients: a=5a = 5, b=−7b = -7, and c=−18c = -18. Now, we'll plug these values into the quadratic formula: $x = \frac-b \pm \sqrt{b^2 - 4ac}}{2a}$ Substituting the values, we get $x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(5)(-18)}2(5)}$ Let's break this down step by step to make sure we don't miss anything. First, we simplify the negative signs $x = \frac{7 \pm \sqrt{(-7)^2 - 4(5)(-18)}10}$ Next, we calculate the terms inside the square root. Remember, a negative number squared becomes positive $x = \frac{7 \pm \sqrt{49 - 4(5)(-18)}10}$ Now, let's multiply −4-4, 55, and −18-18 $x = \frac{7 \pm \sqrt{49 + 360}10}$ Adding 4949 and 360360, we get $x = \frac{7 \pm \sqrt{409}{10}$ So, here we are! We've applied the quadratic formula and simplified it to this point. We have x=7±40910x = \frac{7 \pm \sqrt{409}}{10}. The next step is to analyze this result and see what it tells us about the solutions. The ±\pm sign indicates that we have two possible solutions, one with addition and one with subtraction. The square root of 409409 is not a perfect square, so we'll leave it in radical form for an exact solution. This is perfectly fine and often preferred unless we need an approximate decimal value. Understanding how to handle radicals is crucial in algebra, so make sure you're comfortable with this. The fact that the discriminant, 409409, is positive tells us that we have two distinct real solutions, which aligns with our expectations. Now, let's separate the two solutions and write them out explicitly.

Identifying the Correct Solution

Now that we have x=7±40910x = \frac{7 \pm \sqrt{409}}{10}, let's break down what this means. The ±\pm symbol tells us that there are two possible solutions for xx. One solution comes from adding 409\sqrt{409} to 77, and the other comes from subtracting 409\sqrt{409} from 77. So, we have: Solution 1: $x_1 = \frac7 + \sqrt{409}}{10}$ Solution 2 $x_2 = \frac{7 - \sqrt{409}{10}$ These are the two exact solutions to the quadratic equation 5x2−7x−18=05x^2 - 7x - 18 = 0. Now, let's compare these solutions to the options provided in the question. We had the following options:

A. x=−7±40910x=\frac{-7 \pm \sqrt{409}}{10} B. x=7±40910x=\frac{7 \pm \sqrt{409}}{10} C. x=7±4095x=\frac{7 \pm \sqrt{409}}{5}

By comparing our solutions to the options, we can see that option B, x=7±40910x=\frac{7 \pm \sqrt{409}}{10}, exactly matches what we found. Therefore, option B is the correct solution. It's essential to be meticulous when comparing solutions, especially when dealing with fractions and square roots. A small mistake can lead to selecting the wrong answer. Options A and C are incorrect. Option A has the wrong sign for the 77 in the numerator, and Option C has the wrong denominator. Always double-check your work and compare your results carefully. This process of verification is a key part of problem-solving in mathematics. Once we've identified the correct solution, it's always a good idea to reflect on the problem and the method we used. Did we follow all the steps correctly? Does the solution make sense in the context of the problem? This kind of reflection helps solidify our understanding and improves our problem-solving skills for future challenges.

Conclusion: Mastering Quadratic Equations

Alright, guys, we've reached the end of our journey through solving the quadratic equation 5x2−7x−18=05x^2 - 7x - 18 = 0. We started by understanding what quadratic equations are and how they're represented in general form. Then, we identified the coefficients in our specific equation: a=5a = 5, b=−7b = -7, and c=−18c = -18. After that, we introduced the quadratic formula, a powerful tool for finding the solutions to any quadratic equation. We carefully plugged in the values, step by step, and simplified the expression to x=7±40910x = \frac{7 \pm \sqrt{409}}{10}. We then broke down this result to identify the two solutions: $x_1 = \frac{7 + \sqrt{409}}{10}$

x_2 = \frac{7 - \sqrt{409}}{10}$ By comparing these solutions to the options provided, we confidently selected option B as the correct answer. But more importantly, we've learned the process and the reasoning behind each step. **Understanding quadratic equations** and how to solve them is a fundamental skill in algebra, with applications in various fields. Whether you're dealing with physics problems, engineering designs, or financial calculations, the ability to solve quadratic equations will serve you well. Remember, the key to mastering any mathematical concept is practice. The more you work through problems, the more comfortable and confident you'll become. So, don't hesitate to tackle more quadratic equations and explore different methods of solving them, such as factoring and completing the square. Each method offers a unique perspective and can be useful in different situations. And remember, if you ever get stuck, there are plenty of resources available, including textbooks, online tutorials, and your friendly neighborhood math tutor. Keep practicing, stay curious, and you'll be solving quadratic equations like a pro in no time!