True Equations For X = -2 And X = 2? Find Out!

by Pedro Alvarez 47 views

Hey guys! Today, we're diving into the fascinating world of equations and exploring which ones hold true when we plug in specific values for our variable, x. Specifically, we're going to investigate which of the given equations are satisfied by both x = -2 and x = 2. This is a fundamental concept in algebra, and mastering it will help you tackle more complex problems with confidence. Let's get started and unravel the mysteries of these equations!

Decoding the Equations: A Step-by-Step Approach

To figure out which equations are true for x = -2 and x = 2, we'll take a systematic approach. We'll substitute each value into each equation and see if the equation holds true. Remember, an equation is true if both sides are equal after the substitution. This process might seem a bit tedious, but it's the most reliable way to determine the validity of each equation. So, let's roll up our sleeves and get to work!

Equation 1: $x^2 - 4 = 0$

Let's start with the first equation: $x^2 - 4 = 0$. This equation involves squaring x and then subtracting 4. To check if it's true for x = -2, we substitute -2 for x:

(−2)2−4=4−4=0(-2)^2 - 4 = 4 - 4 = 0

As you can see, the equation holds true for x = -2. Now, let's check for x = 2:

(2)2−4=4−4=0(2)^2 - 4 = 4 - 4 = 0

Again, the equation holds true. This means that $x^2 - 4 = 0$ is a potential candidate for our final answer.

The key concept here is the square of a number. When we square a negative number, we multiply it by itself, resulting in a positive number. This is why (-2)^2 becomes 4. This understanding is crucial for solving various algebraic problems, especially those involving quadratic equations.

Equation 2: $x^2 = -4$

Next up, we have the equation $x^2 = -4$. This equation states that the square of x is equal to -4. Let's substitute x = -2:

(−2)2=4(-2)^2 = 4

Clearly, 4 is not equal to -4, so this equation is not true for x = -2. Now, let's check for x = 2:

(2)2=4(2)^2 = 4

Again, 4 is not equal to -4, so this equation is not true for x = 2 either. We can confidently eliminate this equation from our list.

This equation highlights a critical point: the square of any real number cannot be negative. When you multiply a number by itself, the result is always non-negative (either positive or zero). This fundamental principle is crucial in understanding the nature of square roots and quadratic equations.

Equation 3: $3x^2 + 12 = 0$

Now, let's consider the equation $3x^2 + 12 = 0$. This equation involves squaring x, multiplying it by 3, and then adding 12. Substituting x = -2, we get:

3(−2)2+12=3(4)+12=12+12=243(-2)^2 + 12 = 3(4) + 12 = 12 + 12 = 24

Since 24 is not equal to 0, the equation is not true for x = -2. Let's check for x = 2:

3(2)2+12=3(4)+12=12+12=243(2)^2 + 12 = 3(4) + 12 = 12 + 12 = 24

Again, 24 is not equal to 0, so this equation is not true for x = 2. We can eliminate this equation as well.

This equation demonstrates the importance of the order of operations (PEMDAS/BODMAS). We need to perform the exponentiation (squaring) before the multiplication and addition. Failing to follow the order of operations can lead to incorrect results.

Equation 4: $4x^2 = 16$

Let's move on to the equation $4x^2 = 16$. This equation involves squaring x and then multiplying it by 4. Substituting x = -2, we get:

4(−2)2=4(4)=164(-2)^2 = 4(4) = 16

The equation holds true for x = -2. Now, let's check for x = 2:

4(2)2=4(4)=164(2)^2 = 4(4) = 16

The equation also holds true for x = 2. So, $4x^2 = 16$ is another potential candidate for our answer.

This equation reinforces the concept of solving equations by isolating the variable. We can divide both sides of the equation by 4 to get $x^2 = 4$, which is a simpler form of the equation and makes it easier to see the solutions.

Equation 5: $2(x - 2)^2 = 0$

Finally, let's examine the equation $2(x - 2)^2 = 0$. This equation involves subtracting 2 from x, squaring the result, and then multiplying by 2. Substituting x = -2, we get:

2(−2−2)2=2(−4)2=2(16)=322(-2 - 2)^2 = 2(-4)^2 = 2(16) = 32

Since 32 is not equal to 0, the equation is not true for x = -2. Let's check for x = 2:

2(2−2)2=2(0)2=2(0)=02(2 - 2)^2 = 2(0)^2 = 2(0) = 0

The equation holds true for x = 2, but not for x = -2. Therefore, we can eliminate this equation.

This equation highlights the importance of understanding the properties of zero. Any number multiplied by zero is zero. This principle is crucial for solving equations where one side is equal to zero.

The Verdict: Which Equations Reign Supreme?

After carefully analyzing each equation, we've identified the ones that hold true for both x = -2 and x = 2. Let's recap our findings:

  • Equation 1: $x^2 - 4 = 0$ – This equation is true for both values.
  • Equation 2: $x^2 = -4$ – This equation is false for both values.
  • Equation 3: $3x^2 + 12 = 0$ – This equation is false for both values.
  • Equation 4: $4x^2 = 16$ – This equation is true for both values.
  • Equation 5: $2(x - 2)^2 = 0$ – This equation is true for x = 2, but not for x = -2.

Therefore, the two equations that are true for both x = -2 and x = 2 are:

  • x2−4=0x^2 - 4 = 0

  • 4x2=164x^2 = 16

And there you have it! We've successfully navigated through the world of equations and identified the ones that satisfy our given conditions. Remember, the key to solving these types of problems is to substitute the values carefully and systematically and to understand the fundamental principles of algebra.

Why This Matters: Real-World Applications and Beyond

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