To determine which of the given equations has solutions of 6 and -6, let's analyze each option closely and see which equation satisfies these solutions.
1. Given solutions are 6 and -6. We start by identifying the roots of a quadratic equation.
If an equation has solutions [tex]\( x = 6 \)[/tex] and [tex]\( x = -6 \)[/tex], the quadratic equation can be formed using the fact that [tex]\( x - 6 \)[/tex] and [tex]\( x + 6 \)[/tex] are factors of the equation.
2. Form the equation by multiplying the factors:
[tex]\[
(x - 6)(x + 6) = 0
\][/tex]
3. Expand the equation:
[tex]\[
(x - 6)(x + 6) = x^2 - 36
\][/tex]
So, the quadratic equation that has solutions 6 and -6 is:
[tex]\[
x^2 - 36 = 0
\][/tex]
Now, we will compare this equation with the given options:
1. [tex]\( x^2 - 12x + 36 = 0 \)[/tex]
2. [tex]\( x^2 + 12x - 36 = 0 \)[/tex]
3. [tex]\( x^2 + 36 = 0 \)[/tex]
4. [tex]\( x^2 - 36 = 0 \)[/tex]
Among these options, the equation
[tex]\[
x^2 - 36 = 0
\][/tex]
is the one that has solutions of 6 and -6.
Therefore, the correct equation is:
[tex]\[
\boxed{x^2 - 36 = 0}
\][/tex]
