To find the equation that has the solutions x = 1 + √5 and x = 1 – √5, we can start by rewriting these solutions in a more manageable form. These two values can be represented together as x = 1 ± √5.
We first recognize that we can express these solutions in terms of a quadratic equation. If we let r_1 = 1 + √5 and r_2 = 1 – √5, the general form of a quadratic equation with roots r_1 and r_2 can be written as:
(x – r_1)(x – r_2) = 0.
Replacing r_1 and r_2 with our specific values, we have:
(x – (1 + √5))(x – (1 – √5)) = 0.
Now, we simplify this expression. Expanding the expression gives:
(x – 1 – √5)(x – 1 + √5) = 0
Using the difference of squares formula, this becomes:
(x – 1)² – (√5)² = 0
Which simplifies to:
(x – 1)² – 5 = 0
Now, we can expand (x – 1)²:
x² – 2x + 1 – 5 = 0
Simplifying this gives:
x² – 2x – 4 = 0.
Thus, the quadratic equation that has the solutions x = 1 ± √5 is:
x² – 2x – 4 = 0.