We are asked to solve the quadratic equation $x^2 + x - 1 = 0$ for $x$.

1. Problem Description

We are asked to solve the quadratic equation $x^2 + x - 1 = 0$ for $x$.

2. Solution Steps

We can solve the quadratic equation using the quadratic formula. The general form of a quadratic equation is $ax^2 + bx + c = 0$, and the quadratic formula is:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In our case, we have $a = 1$, $b = 1$, and $c = -1$. Plugging these values into the quadratic formula, we get:

$x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-1)}}{2(1)}$

$x = \frac{-1 \pm \sqrt{1 + 4}}{2}$

$x = \frac{-1 \pm \sqrt{5}}{2}$

Thus, the solutions are $x = \frac{-1 + \sqrt{5}}{2}$ and $x = \frac{-1 - \sqrt{5}}{2}$.

3. Final Answer

The solutions are $x = \frac{-1 + \sqrt{5}}{2}$ and $x = \frac{-1 - \sqrt{5}}{2}$.