Solving 15 + C² = 96 A Step-by-Step Guide To Equations In The Form Ax² = C

JU07/09/2025 08, 2025 by THE IDEN 75 views

Solving Equations in the Form ax² = c: A Detailed Guide to 15 + c² = 96

Introduction

In the realm of mathematics, solving equations is a fundamental skill. This article delves into the process of solving a specific type of equation, one that can be expressed in the form ax² = c. We will use the example equation 15 + c² = 96 to illustrate the steps involved. This type of equation, often encountered in algebra, requires a systematic approach to isolate the variable and find its possible values. Understanding how to manipulate these equations is crucial for various mathematical applications, from basic problem-solving to more advanced concepts in calculus and physics. The ability to solve equations of this form equips students and professionals alike with a powerful tool for tackling real-world problems.

Understanding the Equation Form ax² = c

Before we dive into solving the equation 15 + c² = 96, let's first understand the general form ax² = c. In this form, a and c represent constants, while x is the variable we aim to solve for. The key characteristic of this equation is the presence of the squared variable (x²), which means we are dealing with a quadratic equation, albeit a simplified one. The constant a is the coefficient of the squared term, and c is the constant term. The absence of a linear term (a term with just x) simplifies the solving process, allowing us to isolate x² directly. Recognizing this form is the first step in choosing the appropriate solution method. This type of equation arises frequently in scenarios involving areas, squares, and other geometric shapes, making its understanding essential for practical applications. Moreover, this form serves as a building block for understanding more complex quadratic equations.

Step-by-Step Solution of 15 + c² = 96

Now, let's tackle the equation 15 + c² = 96. Our goal is to rewrite this equation in the form ax² = c, where in our case, x is replaced by c. Here's a step-by-step breakdown:

Isolate the c² term: To begin, we need to isolate the c² term on one side of the equation. This is achieved by subtracting 15 from both sides of the equation:
```
15 + c² - 15 = 96 - 15
c² = 81
```
This step is crucial because it brings the equation closer to the ax² = c form, where a is implicitly 1 in this case.
Take the square root of both sides: Since we have c² = 81, we can find the value of c by taking the square root of both sides. Remember, when taking the square root, we need to consider both positive and negative solutions:
```
√(c²) = ±√81
c = ±9
```
This step is vital as it unveils the two possible solutions for c. It is important to remember the ± sign, as both 9 and -9, when squared, result in 81.
Verify the solutions: To ensure our solutions are correct, we can substitute them back into the original equation:
- For c = 9:
```
15 + (9)² = 15 + 81 = 96
```
- For c = -9:
```
15 + (-9)² = 15 + 81 = 96
```
Both solutions satisfy the original equation, confirming their validity. This verification step is a cornerstone of problem-solving, ensuring accuracy and preventing errors.

The Correct Answer

Based on our step-by-step solution, we find that the correct answer is:

a. c = ±9

This signifies that c can be either 9 or -9, both of which satisfy the given equation. Understanding the dual nature of solutions in quadratic equations is paramount for a comprehensive grasp of algebra. The other options provided are incorrect as they either miss the negative solution or propose incorrect values.

Why Other Options are Incorrect

Let's briefly analyze why the other options are incorrect:

b. c = 9: This option only provides the positive solution and misses the negative solution (-9), which is equally valid.
c. c = ± 979: This is a completely incorrect solution. It seems to be a misunderstanding of the equation's structure and the operations involved.
d. c = 9.79: This is also an incorrect solution. It might stem from a calculation error or a misapplication of a different solving technique.

Identifying why incorrect options are wrong is as important as finding the correct answer. It reinforces understanding and helps prevent similar errors in the future.

Real-World Applications of Equations in the Form ax² = c

Equations in the form ax² = c might seem abstract, but they have numerous real-world applications. Here are a few examples:

Geometry: Calculating the side length of a square given its area. If the area of a square is c and the side length is x, then x² = c.
Physics: Determining the distance an object falls under gravity in a certain time (ignoring air resistance). The equation often involves a squared term for time.
Engineering: Designing structures and calculating stresses and strains, where squared terms often appear in formulas.
Finance: Calculating compound interest or the growth of investments over time.

The ubiquity of this equation form underscores its importance in various disciplines. Understanding how to solve it unlocks the ability to model and analyze a wide array of real-world phenomena.

Tips and Tricks for Solving Equations of this Type

To master solving equations in the form ax² = c, here are some helpful tips and tricks:

Always isolate the squared term first: This is the golden rule. Before taking the square root, ensure that the x² term (or in our case, the c² term) is alone on one side of the equation.
Remember the ± sign: When taking the square root, always consider both positive and negative solutions. This is crucial for obtaining a complete solution.
Simplify radicals: If the square root results in an irrational number, simplify it as much as possible. For example, √8 can be simplified to 2√2.
Check your answers: Substitute your solutions back into the original equation to verify their correctness. This helps prevent errors and builds confidence.
Practice regularly: The more you practice, the more comfortable you will become with solving these types of equations. Work through various examples and exercises.

By adhering to these tips, you can enhance your problem-solving skills and tackle equations with greater ease and accuracy.

Conclusion

In this article, we have explored how to solve equations in the form ax² = c using the example 15 + c² = 96. We've broken down the process into manageable steps, emphasizing the importance of isolating the squared term, considering both positive and negative solutions, and verifying answers. Furthermore, we've highlighted the real-world applications of these equations and offered tips for efficient problem-solving. Mastering this type of equation is a significant step in developing mathematical proficiency and opens doors to understanding more complex concepts in various fields. The ability to confidently solve such equations empowers individuals to tackle challenges both in academic settings and in real-life scenarios. Remember, mathematics is not just about memorizing formulas but about understanding the underlying principles and applying them effectively.