Let's break down and solve the problem step by step to confirm the solution.

The quadratic formula is given as:

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

From the given information:

• $a = 1$,
• $b = -2$,
• $c = -1$.

Substituting these into the formula:

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

Simplify the terms:

1. $-(-2) = 2$,
2. $(-2)^2 = 4$,
3. $-4(1)(-1) = 4$,
4. So, $b^2 - 4ac = 4 + 4 = 8$.

Now substitute: $x = \frac{2 \pm \sqrt{8}}{2}$

Simplify $\sqrt{8}$: $\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}.$

Thus: $x = \frac{2 \pm 2\sqrt{2}}{2}.$

Split the terms:

1. $\frac{2}{2} = 1$,
2. $\frac{2\sqrt{2}}{2} = \sqrt{2}$.

So: $x = 1 \pm \sqrt{2}.$

This provides two solutions: $x_1 = 1 + \sqrt{2}, \quad x_2 = 1 - \sqrt{2}.$

If you are required to enter exact values, ensure you input these expressions correctly: $1 + \sqrt{2}$ or $1 - \sqrt{2}$.

Let me know if you need clarification on any step or further assistance! Here are some related questions for practice:

1. How do you simplify square roots in quadratic equations?
2. What does the discriminant $b^2 - 4ac$ indicate about the solutions?
3. How do you write the final answer if it requires decimal form?
4. How does changing the sign of $b$ affect the quadratic formula?
5. What are the steps for solving a quadratic equation using the formula?

Tip: Always simplify square roots to their simplest radical form before substituting them into the solution.