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

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

Understanding the Equations

Before we begin, let's examine the given equations:

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

We will substitute $x=-2$ and $x=2$ into each equation to determine which ones are true.

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

Let's substitute $x=-2$ and $x=2$ into Equation A:

• For $x=-2$: $(-2)^2-4=4-4=0$
• For $x=2$: $(2)^2-4=4-4=0$

Since both substitutions result in a true statement, Equation A is true for both $x=-2$ and $x=2$.

Equation B: $x^2=-4$

Now, let's substitute $x=-2$ and $x=2$ into Equation B:

• For $x=-2$: $(-2)^2=-4$ is not true
• For $x=2$: $(2)^2=-4$ is not true

Since neither substitution results in a true statement, Equation B is not true for either $x=-2$ or $x=2$.

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

Next, let's substitute $x=-2$ and $x=2$ into Equation C:

• For $x=-2$: $3(-2)^2+12=12+12=24$ is not true
• For $x=2$: $3(2)^2+12=12+12=24$ is not true

Since neither substitution results in a true statement, Equation C is not true for either $x=-2$ or $x=2$.

Equation D: $4x^2=16$

Now, let's substitute $x=-2$ and $x=2$ into Equation D:

• For $x=-2$: $4(-2)^2=16$ is true
• For $x=2$: $4(2)^2=16$ is true

Since both substitutions result in a true statement, Equation D is true for both $x=-2$ and $x=2$.

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

Finally, let's substitute $x=-2$ and $x=2$ into Equation E:

• For $x=-2$: $2(-2-2)^2=2(-4)^2=32$ is not true
• For $x=2$: $2(2-2)^2=2(0)^2=0$ is true

Since only one substitution results in a true statement, Equation E is not true for $x=-2$ but 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. $2(x-2)^2=0$ (for $x=2$)

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

Q: What is the purpose of evaluating equations for specific values of x?

A: Evaluating equations for specific values of x is an essential step in understanding the behavior of the equation. By substituting specific values for the variable, we can determine the truth of the equation and gain insights into its properties.

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

A: To evaluate an equation for a specific value of x, simply substitute the value of x into the equation and simplify the expression. This will give you a true or false statement, indicating whether the equation is true or false for the given value of x.

Q: What if the equation contains variables other than x?

A: If the equation contains variables other than x, you will need to substitute the value of x and any other variables into the equation. For example, if the equation is $y = 2x + 3$, you would substitute the value of x and the value of y into the equation.

Q: Can I use any value of x to evaluate an equation?

A: No, you should only use values of x that are relevant to the problem or equation. For example, if the equation is $x^2 - 4 = 0$, you would only use values of x that are equal to 2 or -2, as these are the only values that satisfy the equation.

Q: How do I know 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, simply substitute the value of x into the equation and simplify the expression. If the resulting statement is true, then the equation is true for the given value of x. If the resulting statement is false, then the equation is false for the given value of x.

Q: Can I use this method to evaluate equations with multiple variables?

A: Yes, you can use this method to evaluate equations with multiple variables. Simply substitute the values of all the variables into the equation and simplify the expression. This will give you a true or false statement, indicating whether the equation is true or false for the given values of the variables.

Q: Are there any limitations to this method?

A: Yes, there are limitations to this method. For example, if the equation contains complex numbers or irrational numbers, you may need to use more advanced mathematical techniques to evaluate the equation. Additionally, if the equation is a polynomial of high degree, you may need to use numerical methods or approximation techniques to evaluate the equation.

Q: Can I use this method to solve equations?

A: Yes, you can use this method to solve equations. By substituting specific values for the variable, you can determine the truth of the equation and gain insights into its properties. This can be a useful technique for solving equations, especially when combined with other mathematical techniques.

Q: Are there any other applications of this method?

A: Yes, there are many other applications of this method. For example, you can use this method to:

• Evaluate the behavior of a function at a specific point
• Determine the maximum or minimum value of a function
• Solve optimization problems
• Evaluate the convergence of a series
• And many other applications

In conclusion, evaluating equations for specific values of x is an essential step in understanding the behavior of the equation. By using this method, you can gain insights into the properties of the equation and solve a wide range of mathematical problems.