Sure, let's examine each equation with the given values [tex]\( x = -2 \)[/tex] and [tex]\( x = 2 \)[/tex] to determine which of them hold true.
1. Equation: [tex]\( x^2 - 4 = 0 \)[/tex]
For [tex]\( x = -2 \)[/tex]:
[tex]\[
(-2)^2 - 4 = 4 - 4 = 0
\][/tex]
So, the equation is true for [tex]\( x = -2 \)[/tex].
For [tex]\( x = 2 \)[/tex]:
[tex]\[
2^2 - 4 = 4 - 4 = 0
\][/tex]
So, the equation is also true for [tex]\( x = 2 \)[/tex].
Therefore, [tex]\( x^2 - 4 = 0 \)[/tex] is true for both [tex]\( x = -2 \)[/tex] and [tex]\( x = 2 \)[/tex].
2. Equation: [tex]\( x^2 = -4 \)[/tex]
For [tex]\( x = -2 \)[/tex]:
[tex]\[
(-2)^2 = 4 \neq -4
\][/tex]
So, the equation is not true for [tex]\( x = -2 \)[/tex].
For [tex]\( x = 2 \)[/tex]:
[tex]\[
2^2 = 4 \neq -4
\][/tex]
So, the equation is also not true for [tex]\( x = 2 \)[/tex].
Therefore, [tex]\( x^2 = -4 \)[/tex] is not true for either [tex]\( x = -2 \)[/tex] or [tex]\( x = 2 \)[/tex].
3. Equation: [tex]\( 3x^2 + 12 = 0 \)[/tex]
For [tex]\( x = -2 \)[/tex]:
[tex]\[
3(-2)^2 + 12 = 3 \cdot 4 + 12 = 12 + 12 = 24 \neq 0
\][/tex]
So, the equation is not true for [tex]\( x = -2 \)[/tex].
For [tex]\( x = 2 \)[/tex]:
[tex]\[
3 \cdot 2^2 + 12 = 3 \cdot 4 + 12 = 12 + 12 = 24 \neq 0
\][/tex]
So, the equation is also not true for [tex]\( x = 2 \)[/tex].
Therefore, [tex]\( 3x^2 + 12 = 0 \)[/tex] is not true for either [tex]\( x = -2 \)[/tex] or [tex]\( x = 2 \)[/tex].
4. Equation: [tex]\( 4x^2 = 16 \)[/tex]
For [tex]\( x = -2 \)[/tex]:
[tex]\[
4(-2)^2 = 4 \cdot 4 = 16
\][/tex]
So, the equation is true for [tex]\( x = -2 \)[/tex].
For [tex]\( x = 2 \)[/tex]:
[tex]\[
4 \cdot 2^2 = 4 \cdot 4 = 16
\][/tex]
So, the equation is also true for [tex]\( x = 2 \)[/tex].
Therefore, [tex]\( 4x^2 = 16 \)[/tex] is true for both [tex]\( x = -2 \)[/tex] and [tex]\( x = 2 \)[/tex].
5. Equation: [tex]\( 2(x-2)^2 = 0 \)[/tex]
For [tex]\( x = -2 \)[/tex]:
[tex]\[
2(-2-2)^2 = 2(-4)^2 = 2 \cdot 16 = 32 \neq 0
\][/tex]
So, the equation is not true for [tex]\( x = -2 \)[/tex].
For [tex]\( x = 2 \)[/tex]:
[tex]\[
2(2-2)^2 = 2 \cdot 0 = 0
\][/tex]
So, the equation is true for [tex]\( x = 2 \)[/tex].
Therefore, [tex]\( 2(x-2)^2 = 0 \)[/tex] is true only for [tex]\( x = 2 \)[/tex].
The equations that are true for the given values [tex]\( x = -2 \)[/tex] and [tex]\( x = 2 \)[/tex] are:
- [tex]\( x^2 - 4 = 0 \)[/tex]
- [tex]\( 4x^2 = 16 \)[/tex]
- [tex]\( 2(x-2)^2 = 0 \)[/tex]
Hence, the correct options are:
- [tex]\( x^2 - 4 = 0 \)[/tex]
- [tex]\( 4x^2 = 16 \)[/tex]
- [tex]\( 2(x-2)^2 = 0 \)[/tex]
