Solving C^2 = 121 A Step-by-Step Guide

Hey guys! Let's dive into solving a classic quadratic equation. We're going to break down the equation $c^2 = 121$ step by step, ensuring you understand not just the solution, but the why behind it. Understanding quadratic equations is super important because they pop up everywhere in math and real-world applications. So, let's get started!

Understanding Quadratic Equations

Before we tackle our specific equation, let's get a handle on what quadratic equations are all about. A quadratic equation is basically any equation that can be written in the general form $ax^2 + bx + c = 0$, where 'a', 'b', and 'c' are constants, and 'a' isn't zero (otherwise, it wouldn't be quadratic anymore!). The solutions to these equations, which are the values of 'x' that make the equation true, are also known as roots or zeros.

Quadratic equations can have up to two real solutions. Think about it – the highest power of 'x' is 2, so we're dealing with a curve (a parabola, to be precise) when we graph it. This curve can intersect the x-axis at two points, one point, or no points at all (which means the solutions are complex numbers, but let’s not go there right now). Different methods, like factoring, completing the square, or using the quadratic formula, are used to find these solutions. Each method has its own strengths and is suitable for different types of quadratic equations. Mastering these methods is key to unlocking the world of quadratic equations and their applications. So, whether you're solving for the trajectory of a ball or optimizing a business process, understanding quadratics is a powerful tool in your mathematical arsenal.

Breaking Down $c^2 = 121$

Now, let's zoom in on our equation: $c^2 = 121$. This equation might look simple, but it's packed with mathematical goodness. The goal here is to find the value(s) of 'c' that, when squared, give us 121. You might immediately think of 11, and you'd be right because $11^2$ (which is 11 times 11) does indeed equal 121. But hold on, there's more to the story!

Remember, we're dealing with squares here, and squaring a negative number gives you a positive number. So, what happens if we square -11? Well, $(-11)^2$ is (-11) times (-11), which also equals 121. This is a crucial point: quadratic equations often have two solutions, and it's easy to miss one if you're not thinking about both positive and negative possibilities. The nature of squaring a number means that both the positive and negative roots of a number will satisfy the equation when squared. This is why it's so important to consider both possibilities when solving equations of this type. Overlooking the negative root is a common mistake, so always double-check to make sure you've captured all possible solutions. Understanding this concept is vital not just for solving this specific equation but for tackling a wide range of quadratic problems in the future. So, keep this in mind as we move forward!

Solving by Taking the Square Root

The most straightforward way to solve $c^2 = 121$ is by taking the square root of both sides. This method directly addresses the squared term and gets us closer to isolating 'c'. Here's how it works:

1. Start with the equation: $c^2 = 121$
2. Take the square root of both sides: $\sqrt{c^2} = \pm\sqrt{121}$

• Notice the '$\pm$' (plus or minus) sign. This is super important! When we take the square root, we need to remember that both the positive and negative roots could be solutions.

3. Simplify: $c = \pm 11$

So, this gives us two possible solutions: $c = 11$ and $c = -11$. Taking the square root of both sides is a powerful technique, but the inclusion of the '$\pm$' sign is crucial to capturing both solutions. This step ensures that we don't inadvertently discard a valid answer, which is a common pitfall in solving quadratic equations. By explicitly considering both positive and negative roots, we maintain the mathematical integrity of our solution and arrive at the complete set of answers. This method is not only efficient but also reinforces the fundamental understanding of how square roots and squaring operations interact in algebraic equations. So, always remember to consider both sides of the coin when dealing with square roots!

Verifying the Solutions

It's always a good idea to check our answers to make sure they're correct. This is especially true in math, where a small mistake can throw everything off. Let's plug our solutions, $c = 11$ and $c = -11$, back into the original equation, $c^2 = 121$, and see if they work.

• For c = 11:

  • Substitute c = 11 into the equation: $(11)^2 = 121$
  • Calculate: $121 = 121$
  • Yep, it checks out!

• For c = -11:

  • Substitute c = -11 into the equation: $(-11)^2 = 121$
  • Calculate: $121 = 121$
  • It works too!

Both solutions satisfy the equation, which gives us confidence that we've found the correct answers. Verifying solutions is a critical step in the problem-solving process. It's not just about getting an answer; it's about ensuring the answer is accurate and valid within the context of the problem. By substituting the solutions back into the original equation, we perform a reality check that confirms our calculations and reasoning. This practice not only helps in identifying and correcting errors but also deepens our understanding of the mathematical concepts involved. So, make it a habit to verify your solutions whenever possible – it's a hallmark of a meticulous and successful problem solver.

Analyzing the Answer Choices

Now, let's look at the answer choices provided and see which ones match our solutions:

A. $c=11$ - This is one of our solutions. B. $c=-11$ - This is also one of our solutions. C. $c=13$ - Let's check: $13^2 = 169$, not 121. So, this is incorrect. D. $c=-13$ - Let's check: $(-13)^2 = 169$, not 121. So, this is also incorrect. E. None of the above - Since we found two correct solutions, this is incorrect.

Therefore, the correct answers are A and B.

Conclusion

So, there you have it! We've successfully solved the equation $c^2 = 121$ and found that the solutions are $c = 11$ and $c = -11$. Remember, when dealing with quadratic equations, always consider both positive and negative roots. Verifying your solutions is a fantastic way to ensure accuracy. Keep practicing, and you'll become a quadratic equation-solving pro in no time! Remember to always double-check your work and understand the underlying concepts, and you'll be well-equipped to tackle more complex mathematical challenges. Happy solving, guys!