To determine which equation has solutions [tex]\( x = 1 \pm \sqrt{5} \)[/tex], we can follow these steps:
1. Identify the form of the solutions:
The given solutions are [tex]\( x = 1 + \sqrt{5} \)[/tex] and [tex]\( x = 1 - \sqrt{5} \)[/tex].
2. Formulate the quadratic equation:
If [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are the solutions of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex], then the equation can be written in the form [tex]\( a(x - p)(x - q) = 0 \)[/tex].
3. Substitute the given solutions into the equation:
Using [tex]\( p = 1 + \sqrt{5} \)[/tex] and [tex]\( q = 1 - \sqrt{5} \)[/tex], we get:
[tex]\[
(x - (1 + \sqrt{5}))(x - (1 - \sqrt{5})) = 0
\][/tex]
4. Expand the equation step-by-step:
[tex]\[
(x - 1 - \sqrt{5})(x - 1 + \sqrt{5}) = 0
\][/tex]
This can be expanded using the difference of squares formula [tex]\( (a - b)(a + b) = a^2 - b^2 \)[/tex]:
[tex]\[
(x - 1)^2 - (\sqrt{5})^2 = 0
\][/tex]
5. Simplify the expression:
Calculate [tex]\( (x - 1)^2 \)[/tex]:
[tex]\[
(x - 1)^2 = x^2 - 2x + 1
\][/tex]
Calculate [tex]\( (\sqrt{5})^2 \)[/tex]:
[tex]\[
(\sqrt{5})^2 = 5
\][/tex]
6. Combine the results:
Putting these together, we have:
[tex]\[
x^2 - 2x + 1 - 5 = 0
\][/tex]
7. Simplify further:
[tex]\[
x^2 - 2x - 4 = 0
\][/tex]
So, the quadratic equation with the solutions [tex]\( x = 1 \pm \sqrt{5} \)[/tex] is:
[tex]\[
x^2 - 2x - 4 = 0
\][/tex]
Therefore, the correct option is:
[tex]\[
\boxed{x^2 - 2x - 4 = 0}
\][/tex]
