Possible Values Of X In (4x-5)^2 = 49
#Exploring the Realm of Quadratic Equations
In the fascinating world of mathematics, quadratic equations hold a special place. They are equations of the form ax^2 + bx + c = 0, where a, b, and c are constants, and x is the variable we seek to find. These equations arise in various fields, from physics and engineering to economics and finance. Understanding how to solve them is crucial for anyone seeking to delve deeper into these disciplines. In this article, we embark on a journey to unravel the solutions for x in the equation (4x-5)^2 = 49, a seemingly simple yet intriguing quadratic equation that holds the key to unlocking valuable insights.
#Unlocking the Secrets of (4x-5)^2 = 49: A Step-by-Step Approach
The equation (4x-5)^2 = 49 presents us with a challenge: to find the values of x that satisfy this equation. To conquer this challenge, we shall employ a strategic approach, breaking down the equation into manageable steps and applying fundamental mathematical principles. Our journey begins with a careful observation: the left-hand side of the equation is a squared term, while the right-hand side is a constant. This observation sparks our first move: taking the square root of both sides of the equation. This maneuver allows us to eliminate the square on the left-hand side, paving the way for a simpler equation to solve.
Taking the square root of both sides, we arrive at: 4x - 5 = ±7. The ± symbol signifies that we must consider both the positive and negative square roots of 49, as both 7 and -7, when squared, yield 49. This crucial step unveils two possible scenarios, each leading to a distinct solution for x. We now have two linear equations to solve: 4x - 5 = 7 and 4x - 5 = -7. Each of these equations can be solved by isolating x, which we will accomplish by performing the same operations on both sides of the equation. This ensures that the equality remains balanced, leading us to the correct solution.
Let's first tackle the equation 4x - 5 = 7. Our goal is to isolate x, and to do so, we must eliminate the -5 on the left-hand side. The natural antidote to subtraction is addition, so we add 5 to both sides of the equation. This yields 4x = 12. Now, x is tantalizingly close to being isolated, but it is still multiplied by 4. The inverse operation of multiplication is division, so we divide both sides of the equation by 4. This final step reveals our first solution: x = 3. This means that when x is 3, the equation (4x-5)^2 = 49 holds true.
Next, we turn our attention to the equation 4x - 5 = -7. We follow the same strategic approach as before, aiming to isolate x. First, we add 5 to both sides of the equation, which gives us 4x = -2. Then, we divide both sides by 4, revealing our second solution: x = -1/2. This means that when x is -1/2, the equation (4x-5)^2 = 49 also holds true. We have now successfully unearthed both solutions to the equation, marking a triumph in our mathematical exploration.
#The Quest for the Correct Answers: A Critical Examination
Having solved the equation, we arrive at two potential solutions: x = 3 and x = -1/2. But our task is not yet complete. We must now critically examine these solutions, comparing them to the options presented and selecting the correct answers. This step ensures that we not only understand the mathematical process but also possess the discernment to identify the accurate solutions within a given context.
Upon careful examination, we find that options A and B perfectly align with our solutions. Option A presents x = -1/2, which we have diligently derived through our calculations. Option B offers x = 3, another solution that we have confidently obtained. These two options stand out as the correct answers, validated by our rigorous mathematical exploration. Options C, D, and E, on the other hand, do not match our solutions and are therefore deemed incorrect. This process of elimination reinforces our understanding and solidifies our confidence in the solutions we have found.
#The Significance of Multiple Solutions: A Deeper Dive
The equation (4x-5)^2 = 49, like many quadratic equations, possesses the intriguing characteristic of having multiple solutions. This arises from the fundamental nature of quadratic equations, where the highest power of the variable is 2. The presence of this squared term allows for two distinct values of x to satisfy the equation. Understanding this concept is crucial for grasping the behavior of quadratic equations and their applications in various fields.
The existence of multiple solutions is not merely a mathematical curiosity; it has profound implications in real-world scenarios. Consider, for instance, the trajectory of a projectile, such as a ball thrown into the air. The height of the ball at any given time can be modeled by a quadratic equation. This equation may have two solutions, representing the two moments when the ball reaches a specific height – once on its way up and once on its way down. Similarly, in financial modeling, quadratic equations can be used to determine the break-even points for a business, where revenue equals costs. These points, often two in number, represent crucial milestones in the business's financial journey.
The concept of multiple solutions extends beyond quadratic equations, permeating various branches of mathematics and its applications. In calculus, for example, finding the roots of a function, where the function's value equals zero, often involves solving equations with multiple solutions. These roots represent critical points on the function's graph, providing valuable information about its behavior. Similarly, in linear algebra, systems of equations can have multiple solutions, representing the points of intersection of lines or planes. These solutions have significant implications in fields such as optimization and computer graphics.
#The Power of Mathematical Exploration: A Concluding Reflection
Our journey through the equation (4x-5)^2 = 49 has been more than just a mathematical exercise; it has been an exploration of fundamental concepts and problem-solving strategies. We have not only unearthed the solutions for x but also delved into the significance of quadratic equations, the importance of multiple solutions, and the broader implications of these concepts in various fields. This journey underscores the power of mathematical exploration, where each equation is a puzzle waiting to be solved, and each solution is a piece of the puzzle that reveals a deeper understanding of the world around us.
#Choosing the Correct Answers: A and B
In conclusion, the possible values of x that satisfy the equation (4x-5)^2 = 49 are -1/2 and 3. Therefore, the correct answers are A and B.
Possible Values of x if (4x-5)^2=49: Choose Two Correct Answers
What are the possible values of x if (4x-5)^2 = 49? This is a classic algebraic problem that involves solving a quadratic equation. We'll walk through the steps to find the correct answers, ensuring a clear understanding of the process. The core of solving this equation lies in recognizing it as a disguised quadratic equation. While it's not in the standard form of ax^2 + bx + c = 0, we can manipulate it to find the solutions for x. Let’s begin by understanding the question and then diving into the solution.
Breaking Down the Equation (4x-5)^2 = 49
To find the possible values of x, we need to understand how to isolate x in the equation (4x-5)^2 = 49. The first crucial step is recognizing that we can take the square root of both sides of the equation. This is a valid operation because if two quantities are equal, their square roots are also equal. However, it’s essential to remember that when taking the square root, we must consider both the positive and negative roots.
Taking the square root of both sides gives us:
4x - 5 = ±7
This splits our problem into two separate equations:
- 4x - 5 = 7
- 4x - 5 = -7
Now we have two simpler linear equations to solve. These linear equations are much easier to handle and will lead us directly to the possible values of x. Solving them involves basic algebraic manipulations that we will tackle step by step.
Solving the First Equation: 4x - 5 = 7
Let's solve the first equation, 4x - 5 = 7. To isolate x, we need to get rid of the -5 on the left side. We can do this by adding 5 to both sides of the equation:
4x - 5 + 5 = 7 + 5
This simplifies to:
4x = 12
Now, to completely isolate x, we need to divide both sides by 4:
4x / 4 = 12 / 4
This gives us our first solution:
x = 3
So, one possible value for x is 3. This is a significant step forward, and we now have one solution. But remember, we have another equation to solve, which may give us a different value for x.
Solving the Second Equation: 4x - 5 = -7
Now let's tackle the second equation, 4x - 5 = -7. Similar to the first equation, our goal is to isolate x. We start by adding 5 to both sides of the equation:
4x - 5 + 5 = -7 + 5
This simplifies to:
4x = -2
Next, we divide both sides by 4 to solve for x:
4x / 4 = -2 / 4
This gives us our second solution:
x = -1/2
So, our second possible value for x is -1/2. We now have two potential solutions for x: 3 and -1/2. It’s crucial to double-check these solutions to ensure they satisfy the original equation.
Verifying the Solutions
To ensure our solutions are correct, we need to substitute them back into the original equation (4x-5)^2 = 49 and see if they hold true.
Verifying x = 3:
(4(3) - 5)^2 = (12 - 5)^2 = (7)^2 = 49
So, x = 3 is indeed a valid solution.
Verifying x = -1/2:
(4(-1/2) - 5)^2 = (-2 - 5)^2 = (-7)^2 = 49
Thus, x = -1/2 is also a valid solution.
Both solutions satisfy the original equation, confirming that we have found the correct values for x. This verification step is a critical part of problem-solving in mathematics, ensuring accuracy and building confidence in our answers.
Identifying the Correct Answer Choices
Now that we have our solutions, x = 3 and x = -1/2, we can match them to the answer choices provided.
Looking at the options:
A. -1/2 B. 3 C. -4/5 D. 5 E. 7
We can see that options A and B match our solutions. Therefore, the two correct answers are A and B. It’s always a good idea to double-check that you’ve selected the correct options before finalizing your answer.
Why Other Options are Incorrect
Understanding why the other options are incorrect is just as important as finding the correct answers. Let's briefly examine why options C, D, and E do not satisfy the original equation.
- C. x = -4/5
(4(-4/5) - 5)^2 = (-16/5 - 25/5)^2 = (-41/5)^2 = 1681/25, which is not equal to 49.
- D. x = 5
(4(5) - 5)^2 = (20 - 5)^2 = (15)^2 = 225, which is not equal to 49.
- E. x = 7
(4(7) - 5)^2 = (28 - 5)^2 = (23)^2 = 529, which is not equal to 49.
These calculations demonstrate that only x = 3 and x = -1/2 make the equation (4x-5)^2 = 49 true.
Key Concepts Revisited
Let's recap the key mathematical concepts we used to solve this problem:
- Square Root Property: The principle of taking the square root of both sides of an equation to eliminate a square. Remembering to consider both positive and negative roots is essential.
- Solving Linear Equations: The ability to isolate the variable x by performing inverse operations (addition, subtraction, multiplication, division) on both sides of the equation.
- Verification: The crucial step of substituting the solutions back into the original equation to ensure they are valid.
These concepts are fundamental in algebra and will be helpful in solving many other types of equations. Practice applying these concepts to build your mathematical problem-solving skills.
Tips for Solving Similar Problems
Here are some tips to help you solve similar algebraic problems:
- Recognize the Form: Identify the type of equation (e.g., quadratic, linear) and the appropriate method to solve it.
- Isolate the Variable: Use inverse operations to isolate the variable you're solving for.
- Consider All Possibilities: When taking square roots, remember to consider both positive and negative roots.
- Verify Your Solutions: Always substitute your solutions back into the original equation to check for accuracy.
- Practice Regularly: The more you practice, the more comfortable and confident you'll become in solving algebraic problems.
Conclusion: Mastering Algebraic Equations
Solving the equation (4x-5)^2 = 49 demonstrates the importance of understanding and applying fundamental algebraic principles. By breaking down the problem into manageable steps, we successfully found the two possible values of x: 3 and -1/2. The correct answer choices are A and B.
Mastering these types of problems builds a strong foundation in algebra and enhances your ability to tackle more complex mathematical challenges. Remember to practice regularly, understand the underlying concepts, and always verify your solutions. With these strategies, you’ll be well-equipped to solve a wide range of algebraic equations.
Therefore, the correct answers are A. -1/2 and B. 3.
