Solving Quadratic Equations: A Step-by-Step Guide

Are you ready to dive into the world of quadratic equations? Don't worry, it's not as scary as it sounds! In this guide, we'll break down how to solve an equation by first transforming it into the form ax² = c⁴. We'll tackle the problem: 15 + c² = 96 and together we'll find the solution, following the methods in a step-by-step manner. Get ready to flex those math muscles and build your problem-solving skills.

Understanding the Basics of Quadratic Equations

First things first, what exactly is a quadratic equation? Simply put, it's an equation that includes a variable raised to the power of two. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. When we talk about solving a quadratic equation, we're looking for the values of the variable (in this case, c) that make the equation true. These values are often called the roots or solutions of the equation. Keep in mind that quadratic equations can have two solutions, one solution, or even no real solutions! The nature of these solutions depends on the values of the constants a, b, and c. The methods used for solving can vary, but the main goal always remains the same: isolating the variable and finding its value(s).

In our particular problem, 15 + c² = 96, it might not look exactly like the standard form ax² + bx + c = 0 at first glance, but with a little rearranging, we can get it into a similar form. The crucial thing to remember is that we want to get the equation into a form where we can easily solve for the variable, c. The process involves several steps, and each one builds upon the last, guiding us closer to the solution. The transformation of our original equation into ax² = c⁴ is a strategic move to simplify the process. This specific form allows us to isolate the squared term, which is the key to solving for c. Understanding this form is like having a secret code, opening a direct path to the solutions. By mastering the core concepts and the algebraic manipulations, you will be able to easily solve this kind of equation.

Solving quadratic equations is a fundamental skill in mathematics, used in many fields, from physics and engineering to economics and computer science. It provides the tools necessary to analyze and model various real-world situations. So, whether you are a student striving to pass an exam or a professional looking to apply mathematical concepts to your work, grasping the fundamentals of solving quadratic equations is invaluable. Furthermore, it helps enhance critical thinking and problem-solving abilities, since, you will be trained to analyze the equation, identify the best solving methods, and apply these methods to reach the correct answer.

Transforming the Equation into the Desired Form

Now, let's get down to the nitty-gritty of solving the equation 15 + c² = 96. Our goal is to rearrange this equation into the form ax² = c⁴. In this case, our x will be equivalent to c. The initial step involves isolating the term with the variable (c²) on one side of the equation. To do this, we need to subtract 15 from both sides of the equation. This gives us c² = 96 - 15. Performing the subtraction simplifies the equation to c² = 81. You can see that we're already on our way to the desired form. We’ve managed to get the variable term by itself, and on the other side we have a constant. Now the equation looks much simpler and it's easier to handle.

Now, we need to consider how this relates to ax² = c⁴. In this specific context, the equation c² = 81 is almost in the desired form, just by using a little trick of algebra. If we rewrite c² as 1 * c²*, you can see that a = 1. This also shows that the result from the other side is actually c⁴, where the power is 4. In our original equation, the power of c is 2, while the result in the equation is equivalent to 81, which is equal to 9². So, our equation is actually in the right form, we just have to compute the final step.

The process of transforming an equation into a particular form might seem like just a series of steps, but it's really all about using algebraic rules and logical reasoning to get the equation in a format that you can easily solve. This kind of manipulation is common in algebra and is essential for being able to solve equations effectively. Always remember that whatever you do to one side of an equation, you must do to the other side as well to keep it balanced. This fundamental principle ensures that you maintain the equality and can validly arrive at the correct answer. The key is to practice, and with enough practice, you’ll become more adept at these transformations, making it a natural part of your problem-solving process.

Solving for the Variable

Now that we have the equation in a simplified form, c² = 81, it's time to solve for c. To do this, we need to get c by itself. The operation that we need to perform is to take the square root of both sides of the equation. Remember, when you take the square root, you must consider both positive and negative solutions. So, when you take the square root of c², you get c = ±√81. This is a crucial step, and often students may forget to include the negative root, leading to an incorrect answer. The plus or minus symbol (±) is a shorthand way of writing “plus or minus,” indicating that the variable can have two possible values.

Taking the square root of 81 gives us 9, so c = ±9. This means the two solutions for the equation are c = 9 and c = -9. These are the values of c that, when plugged back into the original equation, make it a true statement. Always double-check your answers by substituting the values back into the original equation to ensure that they are indeed correct. When you substitute 9 into the original equation 15 + c² = 96, you get 15 + 9² = 15 + 81 = 96. And when you substitute -9 into the equation, you get 15 + (-9)² = 15 + 81 = 96. Both values satisfy the equation.

Therefore, we have successfully solved the equation by first writing it in the form ax² = c⁴. The ability to solve an equation by isolating the variable and finding its value(s) is one of the most important concepts in mathematics, and is a vital step in learning to solve more complex equations. Understanding how to handle both positive and negative solutions is also crucial, as it ensures that you don't miss any valid answers.

Conclusion: Selecting the Correct Answer

Based on our calculations, the correct answer is A. c = ±9. This answer reflects both the positive and negative solutions that we found when we took the square root of 81. Remember that quadratic equations frequently have two solutions, and it is crucial to consider both possibilities. The other options, B, C, and D, are incorrect because they either provide only one solution or approximate the solution without considering the plus/minus sign.

By following these steps, you have successfully solved the equation 15 + c² = 96. You have also learned how to transform the equation into a more manageable form, and understand the importance of considering both positive and negative roots when solving quadratic equations. Now, you can confidently approach similar problems with the knowledge and tools needed to succeed. Practice with different quadratic equations to master this skill. You'll find that with each problem you solve, your understanding and confidence will grow.

Remember, mathematics is a skill that is strengthened with practice. Don't be discouraged if you find it challenging initially; keep practicing, and you'll become more proficient over time. Each problem you solve is an opportunity to strengthen your skills and boost your confidence in solving similar problems.

Now you're well-equipped to tackle quadratic equations. Keep practicing, and you'll find that these equations become more manageable and easier to solve. Good luck, and keep exploring the fascinating world of mathematics!

For more information, visit the Khan Academy website on quadratic equations: Khan Academy - Quadratic Equations.