To find the roots of the function $f(x) = x^2 + 2x - 1$, we solve the equation $x^2 + 2x - 1 = 0$. This is a quadratic equation, and its roots can be determined using the quadratic formula:

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

Here, $a = 1$, $b = 2$, and $c = -1$. Substituting these values into the formula:

$x = \frac{-2 \pm \sqrt{2^2 - 4(1)(-1)}}{2(1)}$ $x = \frac{-2 \pm \sqrt{4 + 4}}{2}$ $x = \frac{-2 \pm \sqrt{8}}{2}$ $x = \frac{-2 \pm 2\sqrt{2}}{2}$ $x = -1 \pm \sqrt{2}$

Thus, the roots of the function are:

$x = -1 + \sqrt{2} \quad \text{and} \quad x = -1 - \sqrt{2}$

Would you like further clarification or visual representation?

Relative Questions:

1. What is the significance of roots in a quadratic function?
2. How does the discriminant determine the nature of the roots?
3. Can you apply the quadratic formula to a different equation, like $x^2 - 4x + 4 = 0$?
4. What are the graphical implications of the roots for $f(x) = x^2 + 2x - 1$?
5. Can we approximate the numerical values of these roots?

Tip:

Always check the discriminant ($b^2 - 4ac$) first to determine if roots are real, complex, or repeated.