Solve The Equation X²=18. Options: A. X= ± 6 √3, B. X= ± 3 √2, C. X= ± 3 √3, D. X= ± 3 √6
In the realm of mathematics, solving equations is a fundamental skill. This article delves into the step-by-step process of finding the solutions to the equation x² = 18. We will explore the underlying mathematical principles and demonstrate how to arrive at the correct answer. This comprehensive guide aims to equip you with the knowledge and confidence to tackle similar problems.
Understanding the Equation
The Basics of Quadratic Equations
At its core, the equation x² = 18 is a quadratic equation, a type of polynomial equation characterized by a variable raised to the power of two. Quadratic equations are ubiquitous in various fields, from physics and engineering to economics and computer science. Understanding how to solve them is crucial for anyone pursuing these disciplines. The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants. In our case, the equation x² = 18 can be rewritten as x² - 18 = 0, fitting the standard form with a = 1, b = 0, and c = -18.
Solving a quadratic equation means finding the values of the variable (in this case, x) that satisfy the equation, also known as the roots or solutions of the equation. Quadratic equations can have up to two distinct real solutions, one repeated real solution, or two complex solutions. The nature and number of solutions depend on the discriminant (b² - 4ac) of the equation. If the discriminant is positive, there are two distinct real solutions; if it is zero, there is one repeated real solution; and if it is negative, there are two complex solutions.
Isolating the Variable
The primary strategy for solving x² = 18 involves isolating the variable x². This is achieved by performing the same operation on both sides of the equation to maintain equality. In this case, the variable x² is already isolated on the left side of the equation, making the next step straightforward. The equation x² = 18 tells us that we are looking for a number that, when multiplied by itself, equals 18. This is where the concept of square roots comes into play.
To isolate x, we need to undo the squaring operation. The inverse operation of squaring is taking the square root. By taking the square root of both sides of the equation, we can eliminate the square on x and solve for its value. However, it is crucial to remember that taking the square root of a number yields both a positive and a negative solution, as both the positive and negative values, when squared, will result in the same positive number. This is a common point of confusion for many students, but it is essential for finding all possible solutions.
Solving for x
Taking the Square Root
To solve the equation x² = 18, we take the square root of both sides. This step is crucial because it reverses the squaring operation, allowing us to isolate x. Remember that when we take the square root of a number, we must consider both the positive and negative roots, as both values, when squared, will yield the original number. Therefore, taking the square root of both sides of x² = 18 gives us:
√x² = ±√18
This simplifies to:
x = ±√18
The ± sign indicates that there are two possible solutions: a positive square root and a negative square root. This is a fundamental concept in algebra and is essential for finding all solutions to quadratic equations. Many students might initially forget to include the negative root, leading to an incomplete answer. Always remember to consider both the positive and negative roots when solving equations involving squares.
Simplifying the Radical
Now we need to simplify the radical expression √18. Simplifying radicals involves expressing the number under the square root in its simplest form by factoring out perfect squares. This makes the radical easier to work with and provides a more elegant solution. The number 18 can be factored into 9 × 2, where 9 is a perfect square (3² = 9). Therefore, we can rewrite √18 as:
√18 = √(9 × 2)
Using the property of square roots that √(a × b) = √a × √b, we can further simplify this as:
√(9 × 2) = √9 × √2
Since √9 = 3, we have:
√9 × √2 = 3√2
Thus, the simplified form of √18 is 3√2. This step is important because it presents the solution in its most reduced and easily understandable form. Simplifying radicals is a skill that is used extensively in algebra and calculus, and mastering it is crucial for success in higher-level mathematics.
The Solutions
Substituting the simplified radical back into our equation, we get:
x = ±3√2
This means there are two solutions for x: x = 3√2 and x = -3√2. These are the values that, when squared, will equal 18. The positive solution, 3√2, is the principal square root of 18, while -3√2 is the negative square root. Understanding that quadratic equations can have two solutions is essential for a complete understanding of algebra.
Therefore, the solutions to the equation x² = 18 are x = 3√2 and x = -3√2. This can be written concisely as x = ±3√2. This result matches option B in the given choices, confirming that it is the correct answer.
Analyzing the Answer Choices
Evaluating the Options
We are given four options for the solutions to the equation x² = 18:
A. x = ±6√3
B. x = ±3√2
C. x = ±3√3
D. x = ±3√6
We have already determined that the correct solutions are x = ±3√2. Let’s analyze why the other options are incorrect.
Option A: x = ±6√3
If x = ±6√3, then x² = (±6√3)² = 36 × 3 = 108, which is not equal to 18. Therefore, this option is incorrect.
Option C: x = ±3√3
If x = ±3√3, then x² = (±3√3)² = 9 × 3 = 27, which is not equal to 18. Therefore, this option is incorrect.
Option D: x = ±3√6
If x = ±3√6, then x² = (±3√6)² = 9 × 6 = 54, which is not equal to 18. Therefore, this option is incorrect.
Correct Answer
Only option B, x = ±3√2, provides the correct solutions. When we square ±3√2, we get:
(±3√2)² = 9 × 2 = 18
This confirms that option B is the correct answer. Analyzing the incorrect options helps to reinforce the understanding of why the correct solution is what it is. By squaring the proposed solutions and checking if they equal 18, we can easily eliminate the incorrect choices.
Common Mistakes and How to Avoid Them
Forgetting the Negative Root
One of the most common mistakes when solving quadratic equations is forgetting to include both the positive and negative roots. When taking the square root of a number, it is crucial to remember that both a positive and a negative value, when squared, will result in the same positive number. For example, both (3√2)² and (-3√2)² equal 18. To avoid this mistake, always include the ± sign when taking the square root of both sides of an equation.
Incorrectly Simplifying Radicals
Another common error is incorrectly simplifying radicals. This can happen if you don’t correctly factor the number under the square root or if you apply the properties of square roots incorrectly. To avoid this, practice simplifying radicals regularly and double-check your work. Remember that you can only simplify a square root if the number under the root has a perfect square factor. For instance, √18 can be simplified because 18 has a perfect square factor of 9.
Arithmetic Errors
Simple arithmetic errors can also lead to incorrect solutions. This can include mistakes in squaring numbers, multiplying, or adding. To minimize these errors, work carefully and double-check each step. Use a calculator if necessary to verify your calculations. Paying attention to detail is key to avoiding arithmetic errors.
Not Checking the Solutions
Finally, failing to check the solutions is a common mistake. After solving an equation, it is always a good practice to plug the solutions back into the original equation to ensure they are correct. This step can help you catch any errors you might have made and give you confidence in your answer. In our case, substituting ±3√2 back into x² = 18 confirms that these are indeed the correct solutions.
Real-World Applications
Practical Examples
Quadratic equations are not just abstract mathematical concepts; they have numerous real-world applications. One common application is in physics, where quadratic equations are used to describe projectile motion. For example, the height of a ball thrown into the air can be modeled using a quadratic equation. Solving this equation can tell you how long it will take for the ball to hit the ground or what its maximum height will be.
Another application is in engineering, where quadratic equations are used in the design of structures and systems. For instance, the shape of a parabolic arch, which is often used in bridges, can be described by a quadratic equation. Engineers need to solve these equations to ensure the structural integrity of the bridge.
Quadratic equations also appear in economics, where they can be used to model cost and revenue functions. Businesses can use these models to determine the optimal price for their products or the quantity they should produce to maximize profit. Understanding and solving quadratic equations is therefore a valuable skill in various professional fields.
Importance of Problem-Solving Skills
Solving equations like x² = 18 not only teaches us how to manipulate mathematical expressions but also enhances our problem-solving skills. These skills are transferable and beneficial in many areas of life. The process of identifying the problem, developing a strategy, executing the steps, and checking the solution is a valuable skill that can be applied to a wide range of challenges.
In addition, mastering quadratic equations lays a solid foundation for more advanced mathematical concepts, such as calculus and differential equations. These concepts are crucial in many scientific and engineering disciplines. Therefore, a thorough understanding of quadratic equations is essential for anyone pursuing a career in these fields.
Conclusion
Recap of the Solution
In this article, we have explored the process of solving the equation x² = 18. We started by understanding the basics of quadratic equations and the importance of isolating the variable. We then took the square root of both sides of the equation, remembering to include both the positive and negative roots. We simplified the radical expression and arrived at the solutions x = ±3√2.
We also analyzed the given answer choices, demonstrating why the other options were incorrect. We discussed common mistakes that students make when solving quadratic equations and provided tips on how to avoid them. Finally, we highlighted the real-world applications of quadratic equations and the importance of problem-solving skills.
Final Thoughts
Solving equations is a fundamental skill in mathematics, and mastering quadratic equations is a crucial step in building a strong mathematical foundation. By understanding the underlying principles and practicing regularly, you can confidently tackle these types of problems. Remember to always consider both positive and negative roots, simplify radicals correctly, and check your solutions.
This comprehensive guide has provided you with the knowledge and tools to solve the equation x² = 18 and similar problems. Keep practicing and exploring the world of mathematics, and you will continue to develop your problem-solving skills and deepen your understanding of the world around you.