Which Equations Are True For $x = -2$ And $x = 2$? Select Two Options.A. $x^2 - 4 = 0$ B. \$x^2 = -4$[/tex\] C. $3x^2 + 12 = 0$ D. $4x^2 = 16$ E. $2(x - 2)^2 = 0$

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In mathematics, equations are used to represent relationships between variables. When evaluating equations, it's essential to consider the values of the variables involved. In this article, we will explore which equations are true for specific values of x, namely x = -2 and x = 2.

Understanding the Equations

Before we dive into the evaluation process, let's take a closer look at the equations provided:

  • A. $x^2 - 4 = 0$
  • B. $x^2 = -4$
  • C. $3x^2 + 12 = 0$
  • D. $4x^2 = 16$
  • E. $2(x - 2)^2 = 0$

Evaluating Equation A: $x^2 - 4 = 0$

To evaluate equation A, we need to substitute x = -2 and x = 2 into the equation.

For x = -2:

(βˆ’2)2βˆ’4=4βˆ’4=0(-2)^2 - 4 = 4 - 4 = 0

The equation is true for x = -2.

For x = 2:

(2)2βˆ’4=4βˆ’4=0(2)^2 - 4 = 4 - 4 = 0

The equation is also true for x = 2.

Evaluating Equation B: $x^2 = -4$

For x = -2:

(βˆ’2)2=4β‰ βˆ’4(-2)^2 = 4 \neq -4

The equation is not true for x = -2.

For x = 2:

(2)2=4β‰ βˆ’4(2)^2 = 4 \neq -4

The equation is not true for x = 2.

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

For x = -2:

3(βˆ’2)2+12=12+12=24β‰ 03(-2)^2 + 12 = 12 + 12 = 24 \neq 0

The equation is not true for x = -2.

For x = 2:

3(2)2+12=12+12=24β‰ 03(2)^2 + 12 = 12 + 12 = 24 \neq 0

The equation is not true for x = 2.

Evaluating Equation D: $4x^2 = 16$

For x = -2:

4(βˆ’2)2=164(-2)^2 = 16

The equation is true for x = -2.

For x = 2:

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

The equation is also true for x = 2.

Evaluating Equation E: $2(x - 2)^2 = 0$

For x = -2:

2(βˆ’2βˆ’2)2=2(βˆ’4)2=32β‰ 02(-2 - 2)^2 = 2(-4)^2 = 32 \neq 0

The equation is not true for x = -2.

For x = 2:

2(2βˆ’2)2=2(0)2=02(2 - 2)^2 = 2(0)^2 = 0

The equation is true for x = 2.

Conclusion

In conclusion, the equations that are true for x = -2 and x = 2 are:

  • A. $x^2 - 4 = 0$
  • D. $4x^2 = 16$
  • E. $2(x - 2)^2 = 0$

In this article, we will address some of the most frequently asked questions related to the evaluation of equations for specific values of x.

Q: What is the difference between an equation and an expression?

A: An equation is a statement that expresses the equality of two mathematical expressions, while an expression is a combination of variables, constants, and mathematical operations.

Q: How do I evaluate an equation for a specific value of x?

A: To evaluate an equation for a specific value of x, substitute the value of x into the equation and simplify the resulting expression.

Q: What is the order of operations when evaluating an equation?

A: The order of operations when evaluating an equation is:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I determine if an equation is true or false for a specific value of x?

A: To determine if an equation is true or false for a specific value of x, substitute the value of x into the equation and simplify the resulting expression. If the resulting expression is equal to zero, the equation is true. If the resulting expression is not equal to zero, the equation is false.

Q: What is the significance of the value of x in an equation?

A: The value of x in an equation represents the input or independent variable. The value of x can affect the output or dependent variable of the equation.

Q: How do I choose the correct value of x to evaluate an equation?

A: When choosing the correct value of x to evaluate an equation, consider the context of the problem and the specific requirements of the equation. In some cases, it may be necessary to evaluate the equation for multiple values of x to determine the behavior of the equation.

Q: What are some common mistakes to avoid when evaluating equations?

A: Some common mistakes to avoid when evaluating equations include:

  • Not following the order of operations
  • Not simplifying the resulting expression
  • Not considering the context of the problem
  • Not evaluating the equation for multiple values of x

Q: How do I use equations in real-world applications?

A: Equations are used in a wide range of real-world applications, including:

  • Physics and engineering: Equations are used to describe the motion of objects, the behavior of electrical circuits, and the properties of materials.
  • Economics: Equations are used to model economic systems, predict economic trends, and make informed decisions.
  • Computer science: Equations are used to develop algorithms, model complex systems, and optimize performance.

Conclusion

In conclusion, evaluating equations for specific values of x is a fundamental concept in mathematics. By understanding the order of operations, simplifying expressions, and considering the context of the problem, you can effectively evaluate equations and make informed decisions in a wide range of real-world applications.