Equations True For X=-2 And X=2? Find The Solutions!

Hey guys! Today, we're diving into a fun math problem where we need to figure out which equations are true when x is either -2 or 2. It might sound a bit tricky, but don't worry, we'll break it down step by step. We have five equations, and our mission is to select the two that work perfectly for both x values. Let's get started!

Understanding the Problem

Before we jump into solving the equations, let's make sure we understand what we're looking for. We're given two values for x: -2 and 2. An equation is "true" for a specific value of x if, after substituting that value into the equation, both sides of the equation are equal. For example, if we have the equation x + 3 = 5, and we substitute x = 2, we get 2 + 3 = 5, which simplifies to 5 = 5. Since both sides are equal, the equation is true for x = 2. Our task is to find two equations from the list that are true for both x = -2 and x = 2. This means we'll need to test each equation with both values to see if they hold up. Remember, mathematical equations are like puzzles; each part needs to fit perfectly for the solution to be correct. Think of it as a balancing act, where both sides of the equation must remain equal. Substituting the values of x is like placing weights on a scale; if the scale balances, the equation is true for that value of x. So, we're looking for the equations where this balancing act works for both -2 and 2. It's also important to understand the basic operations involved, such as squaring a number. Squaring a number means multiplying it by itself. For example, 2^2 (2 squared) is 2 * 2 = 4, and (-2)^2 (-2 squared) is (-2) * (-2) = 4. Notice that squaring a negative number results in a positive number, which is a crucial concept for this problem. Another key thing to remember is the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order tells us which operations to perform first when simplifying an expression. For example, in the equation 3x^2 + 12 = 0, we need to square x first, then multiply by 3, and finally add 12. Keeping these concepts in mind will make solving the equations much easier and more straightforward. We're not just plugging in numbers; we're understanding how these numbers interact within the mathematical framework of each equation. Now, let's put these concepts into action and start testing the equations.

Testing the Equations

Okay, let's dive into testing each equation one by one. We'll substitute x = -2 and x = 2 into each equation and see if it holds true. Remember, for an equation to be true, both sides must be equal after the substitution.

Option 1: x² - 4 = 0

Let's start with the first option: x² - 4 = 0. First, we'll substitute x = -2:

(-2)² - 4 = 0

4 - 4 = 0

0 = 0

So, the equation holds true for x = -2. Now, let's try x = 2:

(2)² - 4 = 0

4 - 4 = 0

0 = 0

Great! The equation also holds true for x = 2. This equation is a strong contender, so let's keep it in mind as we move forward. Testing equations systematically like this is key to ensuring accuracy. Each step is a small piece of the puzzle, and by carefully working through each one, we can confidently arrive at the correct solution. Think of it like detective work, where each substitution is a clue that brings us closer to solving the mystery. The beauty of algebra is that it provides a clear and logical framework for solving problems, and this process of substitution is a fundamental tool in that framework.

Option 2: x² = -4

Next up, we have the equation x² = -4. Let's substitute x = -2:

(-2)² = -4

4 = -4

This is not true, so the equation fails for x = -2. Since the equation needs to be true for both values of x, we can already eliminate this option. It's important to recognize when an equation fails early in the testing process, as it saves time and effort. This highlights a key principle in problem-solving: efficiency. By quickly identifying and discarding incorrect options, we can focus our attention on the more promising candidates. In this case, the square of any real number cannot be negative. This is a fundamental concept in algebra, and recognizing it can help us quickly eliminate options. Therefore, without even testing x = 2, we know this equation is not a solution. It's like knowing a vital piece of information that immediately rules out a suspect in a detective case. This kind of pattern recognition is a valuable skill in mathematics and problem-solving in general.

Option 3: 3x² + 12 = 0

Now, let's test the equation 3x² + 12 = 0. Substitute x = -2:

3(-2)² + 12 = 0

3(4) + 12 = 0

12 + 12 = 0

24 = 0

This is not true, so the equation fails for x = -2. Again, we can eliminate this option. This equation presents another important lesson in mathematical reasoning: understanding the implications of each term. In this case, 3x² will always be a non-negative value (since squaring any real number results in a non-negative value, and multiplying by 3 keeps it non-negative). Adding 12 to a non-negative value will always result in a positive value, so the expression can never equal 0. This kind of insight can save us from lengthy calculations and helps us develop a deeper understanding of the underlying principles at play.

Option 4: 4x² = 16

Let's move on to 4x² = 16. Substituting x = -2:

4(-2)² = 16

4(4) = 16

16 = 16

This equation holds true for x = -2. Now, let's try x = 2:

4(2)² = 16

4(4) = 16

16 = 16

The equation also holds true for x = 2! This looks like our second solution.

Option 5: 2(x - 2)² = 0

Finally, let's test 2(x - 2)² = 0. Substituting x = -2:

2(-2 - 2)² = 0

2(-4)² = 0

2(16) = 0

32 = 0

This is not true, so the equation fails for x = -2, and we can eliminate it.

Determining the Solutions

Alright, we've tested all the equations, and it's time to identify the two that are true for both x = -2 and x = 2. Remember, we were looking for equations where substituting either -2 or 2 for x would result in a true statement.

After carefully working through each option, we found that:

• x² - 4 = 0 holds true for both x = -2 and x = 2.
• 4x² = 16 also holds true for both x = -2 and x = 2.

So, these are our two solutions! Successfully identifying these solutions demonstrates the importance of systematic testing and careful substitution in algebra. We didn't just guess; we followed a logical process, plugging in the values and verifying whether the equations held true. This method is not only effective but also builds a solid foundation for more advanced mathematical concepts. It's like mastering the fundamentals of a sport before moving on to more complex strategies. By understanding how to solve basic equations, we pave the way for tackling more challenging problems in the future. Furthermore, this process reinforces the idea that mathematics is not about memorizing formulas, but about understanding the relationships between numbers and operations. Each equation is a story, and by substituting the values, we're uncovering whether the story makes sense. This approach to problem-solving is applicable not only in mathematics but also in various aspects of life, where critical thinking and logical reasoning are essential. Now that we've found our solutions, let's summarize what we've learned and reinforce the key concepts.

Final Answer

So, after testing all the equations, we found that the two equations that hold true for both x = -2 and x = 2 are:

• x² - 4 = 0
• 4x² = 16

That's it, guys! We've successfully solved the problem. Remember, the key to solving these types of questions is to carefully substitute the given values and check if the equation holds true. Math can be fun when you approach it step by step! This exercise wasn't just about finding the correct answers; it was about developing a problem-solving strategy that can be applied to various mathematical challenges. By systematically testing each option, we ensured that we didn't miss any potential solutions and that we arrived at the correct answer with confidence. This approach can be particularly useful in exams or situations where accuracy is paramount. Moreover, understanding why certain equations work and others don't is crucial for building a deeper understanding of algebra. For instance, we saw how the square of a number can't be negative, which helped us eliminate one of the options quickly. These kinds of insights come from practice and a willingness to explore the underlying principles behind the equations. As you continue your mathematical journey, remember that consistency and a step-by-step approach will be your best allies. Keep practicing, keep exploring, and you'll find that even the most challenging problems can be broken down into manageable steps. And most importantly, don't be afraid to ask questions and seek help when you need it. Learning is a collaborative process, and together, we can all conquer the world of mathematics! 🚀