The Regula Falsi method (also known as the False Position method) is used to find the real root of a given function within an interval. The function provided is:

$f(x) = x^2 + x - 1$

We are asked to find the root of this equation within the interval $[1, 2]$. The steps for the Regula Falsi method are as follows:

Step 1: Check values at $a = 1$ and $b = 2$

First, evaluate the function at the endpoints $a = 1$ and $b = 2$:

$f(1) = 1^2 + 1 - 1 = 1$ $f(2) = 2^2 + 2 - 1 = 5$

Since $f(1) = 1$ and $f(2) = 5$, the signs are not opposite, which suggests that the interval $[1, 2]$ does not bracket the root, but we can continue to calculate further to refine this interval.

However, Regula Falsi needs a bracketing interval (i.e., $f(a) \times f(b) < 0$). Let's instead check a value between 1 and 2 to adjust the interval or refine the next point.

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