Final answer:
The equation with the solutions x = 1 ± √5 is x² - 2x - 4 = 0. This is determined by using the quadratic formula and matching the constant and linear coefficients to the product and sum of the roots, respectively.
Step-by-step explanation:
The equation that has the solutions x = 1 ± √5 is the one where, when expanded, it will create a quadratic equation that matches these solutions. To find which of the given equations matches, we can either use the quadratic formula directly on the provided equations or try to factor them to match the solutions.
Recall that the quadratic formula is x = (-b ± √(b² - 4ac)) / 2a for a quadratic equation of the form ax² + bx + c = 0. Therefore, by using the given solutions, we can reverse-engineer the equation. The equation resulting in roots 1 + √5 and 1 - √5 must have a constant term that is equal to the product of these roots and a linear coefficient that is the negative sum of the roots.
The correct equation that yields these solutions is x² - 2x - 4 = 0. This is because (1 + √5)(1 - √5) equals -4 (the product of the roots and the constant term) and -(1 + √5) - (1 - √5) simplifies to -2 (the coefficient of the linear term).