The solutions x = 1 ± √5 indicate that we have two distinct values for x: one is x = 1 + √5 and the other is x = 1 – √5. To find the quadratic equation that corresponds to these solutions, we can start from the general property of roots in a quadratic equation, which can be expressed in the form:
x² – (sum of roots)x + (product of roots) = 0
First, let’s calculate the sum and the product of the roots:
- Sum of roots: (1 + √5) + (1 – √5) = 2
- Product of roots: (1 + √5)(1 – √5) = 1² – (√5)² = 1 – 5 = -4
Now substituting these values into the quadratic equation format:
x² – (sum of roots)x + (product of roots) = 0
We get:
x² – 2x – 4 = 0
This is the quadratic equation whose solutions are x = 1 ± √5. Therefore, if you encounter the solutions 1 ± √5, you can confidently say that the equation is x² – 2x – 4 = 0.